{"id":120,"date":"2021-06-02T01:52:21","date_gmt":"2021-06-02T01:52:21","guid":{"rendered":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/1-3\/"},"modified":"2023-11-01T17:22:00","modified_gmt":"2023-11-01T17:22:00","slug":"1-3","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/1-3\/","title":{"raw":"1.2 Integers","rendered":"1.2 Integers"},"content":{"raw":"[latexpage]\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nBy the end of this section it is expected that you will be able to:\r\n<ul>\r\n \t<li>Use negatives and opposites<\/li>\r\n \t<li>Simplify: expressions with absolute value<\/li>\r\n \t<li>Add ans subtract integers<\/li>\r\n \t<li>Multiply and divide integers<\/li>\r\n \t<li>Simplify Expressions with Integers<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<h1>Use Negatives and Opposites<\/h1>\r\n<p id=\"fs-id1170655222181\">If you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. <strong data-effect=\"bold\">Negative numbers<\/strong> are numbers less than \\(0\\). The <span class=\"no-emphasis\" data-type=\"term\">negative numbers<\/span> are to the left of zero on the number line. See <a class=\"autogenerated-content\" href=\"#figure1\">Figure 1<\/a>.<a id=\"figure1\"><\/a><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"389\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_001_new.jpg\" alt=\"A number line extends from negative 4 to 4. A bracket is under the values \u201cnegative 4\u201d to \u201c0\u201d and is labeled \u201cNegative numbers\u201d. Another bracket is under the values 0 to 4 and labeled \u201cpositive numbers\u201d. There is an arrow in between both brackets pointing upward to zero.\" width=\"389\" height=\"113\" data-media-type=\"image\/jpeg\" \/> Figure 1 The number line shows the location of positive and negative numbers.[\/caption]\r\n\r\nThe arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number, and there is no smallest negative number.\r\n<p id=\"fs-id1170655007025\">Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative.<\/p>\r\nConsider how numbers are ordered on the number line. Going from left to right, the numbers increase in value. Going from right to left, the numbers decrease in value. See <a class=\"autogenerated-content\" href=\"#figure2\">Figure 2<\/a>.<a id=\"figure2\"><\/a>\r\n<div id=\"CNX_ElemAlg_Figure_01_03_002_new\" class=\"bc-figure figure\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"440\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_002_new.jpg\" alt=\"A number line ranges from negative 4 to 4. An arrow above the number line extends from negative 1 towards 4 and is labeled \u201clarger\u201d. An arrow below the number line extends from 1 towards negative 4 and is labeled \u201csmaller\u201d.\" width=\"440\" height=\"102\" data-media-type=\"image\/jpeg\" \/> Figure 2 The numbers on a number line increase in value going from left to right and decrease in value going from right to left.[\/caption]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\nRemember that we use the notation:\r\n<p id=\"fs-id1170654965355\"><em data-effect=\"italics\">a<\/em> &lt; <em data-effect=\"italics\">b<\/em> (read \u201ca is less than b\u201d) when a is to the left of b on the number line.<\/p>\r\n<p id=\"fs-id1170654901053\"><em data-effect=\"italics\">a<\/em> &gt; <em data-effect=\"italics\">b<\/em> (read \u201c<em data-effect=\"italics\">a<\/em> is greater than <em data-effect=\"italics\">b<\/em>\u201d) when <em data-effect=\"italics\">a<\/em> is to the right of <em data-effect=\"italics\">b<\/em> on the number line.<\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1170654965087\">Now we need to extend the number line which showed the whole numbers to include negative numbers, too. The numbers marked by points in <a class=\"autogenerated-content\" href=\"#figure3\">Figure 3<\/a> are called the integers. The integers are the numbers \\(\\text{\u2026}-3,-2,-1,0,1,2,3\\text{\u2026}\\)<a id=\"figure3\"><\/a><\/p>\r\n\r\n<div id=\"CNX_ElemAlg_Figure_01_03_003_new\" class=\"bc-figure figure\">\r\n<div class=\"bc-figcaption figcaption\"><\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"401\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_003_new.jpg\" alt=\"A number line extends from negative four to four. Points are plotted at negative four, negative three, negative two, negative one, zero, one, two, 3, and four.\" width=\"401\" height=\"44\" data-media-type=\"image\/jpeg\" \/> Figure 3 All the marked numbers are called integers.[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655160945\" data-type=\"problem\">\r\n<p id=\"fs-id1170654952892\">Order each of the following pairs of numbers, using &lt; or &gt;: a) \\(14\\)___\\(6\\) b) \\(-1\\)___\\(9\\) c) \\(-1\\)___\\(-4\\) d) \\(2\\)___\\(-20\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655177826\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<p id=\"fs-id1170655192048\">It may be helpful to refer to the number line shown.<\/p>\r\n<span id=\"fs-id1170655201463\" data-type=\"media\" data-alt=\"A number line ranges from negative twenty to fifteen with ticks marks between numbers. Every fifth tick mark is labeled a number. Points are plotted at points negative twenty, negative 4, negative 1, 2, 6, 9 and 14.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_004_img_new.jpg\" alt=\"A number line ranges from negative twenty to fifteen with ticks marks between numbers. Every fifth tick mark is labeled a number. Points are plotted at points negative twenty, negative 4, negative 1, 2, 6, 9 and 14.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<table id=\"eip-672\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>a) 14 is to the right of 6 on the number line.<\/td>\r\n<td>\\(14\\)___\\(6\\)\u00a0 \\(14\\) &gt; \\(6\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>b) \u22121 is to the left of 9 on the number line.<\/td>\r\n<td>\\(-1\\)___\\(9\\)\u00a0 \\(-1&lt;9\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>c) \u22121 is to the right of \u22124 on the number line.<\/td>\r\n<td>\\(-1\\)___\\(-4\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>d) 2 is to the right of \u221220 on the number line.<\/td>\r\n<td>\\(2\\)___\\(-20\\) \\(2\\) &gt; \\(-20\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655162060\" data-type=\"problem\">\r\n<p id=\"fs-id1170654967573\">Order each of the following pairs of numbers, using &lt; or &gt; \\(\\text{:}\\) a) \\(15\\)___\\(7\\) b) \\(-2\\)___\\(5\\) c) \\(-3\\)___\\(-7\\) d) \\(5\\)___\\(-17\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654978104\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654967500\">a) &gt; b) &lt; c) &gt; d) &gt;<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170655114791\">You may have noticed that, on the <span class=\"no-emphasis\" data-type=\"term\">number line<\/span>, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers 2 and \\(-2\\) are the same distance from zero, they are called opposite<strong data-effect=\"bold\">s<\/strong>. The opposite of 2 is \\(-2\\), and the opposite of \\(-2\\) is 2<\/p>\r\n\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Opposite<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe <strong data-effect=\"bold\">opposite<\/strong> of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654959734\"><a class=\"autogenerated-content\" href=\"#figure4\">(Figure 4)<\/a> illustrates the definition.<\/p>\r\nThe opposite of 3 is \\(-3\\).<a id=\"figure4\"><\/a>\r\n<div id=\"CNX_ElemAlg_Figure_01_03_005_new\" class=\"bc-figure figure\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"440\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_005_new.jpg\" alt=\"A number line ranges from negative 4 to 4. There are two brackets above the number line. The bracket on the left spans from negative three to 0. The bracket on the right spans from zero to three. Points are plotted on both negative three and three.\" width=\"440\" height=\"81\" data-media-type=\"image\/jpeg\" \/> Figure 4[\/caption]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div id=\"fs-id1170655189167\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Opposite Notation<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1170655174724\">\\(-a\\) means the opposite of the number <em data-effect=\"italics\">a<\/em>.<\/p>\r\n<p id=\"fs-id1170655160739\">The notation \\(-a\\) is read as \u201cthe opposite of <em data-effect=\"italics\">a<\/em>.\u201d<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654920902\">The whole numbers and their opposites are called the integers. The integers are the numbers \\(\\text{\u2026}-3,-2,-1,0,1,2,3\\text{\u2026}\\)<\/p>\r\n\r\n<div id=\"fs-id1170655134778\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Integers<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1170654954489\">The whole numbers and their opposites are called the <strong data-effect=\"bold\">integers<\/strong>.<\/p>\r\n<p id=\"fs-id1170654878555\">The integers are the numbers<\/p>\r\n\r\n<div id=\"fs-id1170655114692\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\text{\u2026}-3,-2,-1,0,1,2,3\\text{\u2026}\\)<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Simplify: Expressions with Absolute Value<\/h1>\r\nWe saw that numbers such as \\(3\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}-3\\) are opposites because they are the same distance from 0 on the number line. They are both two units from 0. The distance between 0 and any number on the number line is called the <strong data-effect=\"bold\">absolute value<\/strong> of that number.\r\n<div id=\"fs-id1170655195725\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Absolute Value<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1170655354299\">The absolute value of a number is its distance from 0 on the number line.<\/p>\r\n<p id=\"fs-id1170654982157\">The absolute value of a number <em data-effect=\"italics\">n<\/em> is written as \\(|n|\\).<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170655160915\">For example,<\/p>\r\n\r\n<ul id=\"fs-id1166424918316\" data-bullet-style=\"bullet\">\r\n \t<li>\\(-5\\phantom{\\rule{0.2em}{0ex}}\\text{is}\\phantom{\\rule{0.2em}{0ex}}5\\) units away from \\(0\\), so \\(|-5|=5\\).<\/li>\r\n \t<li>\\(\\phantom{\\rule{0.65em}{0ex}}5\\phantom{\\rule{0.2em}{0ex}}\\text{is}\\phantom{\\rule{0.2em}{0ex}}5\\) units away from \\(0\\), so \\(|5|=5\\).<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1170655134144\"><a class=\"autogenerated-content\" href=\"#figure5\">Figure 5<\/a> illustrates this idea.<\/p>\r\nThe integers \\(5\\phantom{\\rule{0.2em}{0ex}}\\text{and are}\\phantom{\\rule{0.2em}{0ex}}5\\) units away from \\(0\\).<a id=\"figure5\"><\/a>\r\n<div id=\"CNX_ElemAlg_Figure_01_03_009_new\" class=\"bc-figure figure\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"385\"]<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_009_new.jpg\" alt=\"A number line is shown ranging from negative 5 to 5. A bracket labeled \u201c5 units\u201d lies above the points negative 5 to 0. An arrow labeled \u201cnegative 5 is 5 units from 0, so absolute value of negative 5 equals 5.\u201d is written above the labeled bracket. A bracket labeled \u201c5 units\u201d lies above the points \u201c0\u201d to \u201c5\u201d. An arrow labeled \u201c5 is 5 units from 0, so absolute value of 5 equals 5.\u201d and is written above the labeled bracket.\" width=\"385\" height=\"130\" data-media-type=\"image\/jpeg\" \/> Figure 5[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1170654957170\">The absolute value of a number is never negative (because distance cannot be negative). The only number with absolute value equal to zero is the number zero itself, because the distance from \\(0\\phantom{\\rule{0.2em}{0ex}}\\text{to}\\phantom{\\rule{0.2em}{0ex}}0\\) on the number line is zero units.<\/p>\r\n\r\n<div id=\"fs-id1170654997376\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Property of Absolute Value<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1170654916082\">\\(|n|\\ge 0\\) for all numbers<\/p>\r\n<p id=\"fs-id1170655074945\">Absolute values are always greater than or equal to zero!<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170655150245\">Mathematicians say it more precisely, \u201cabsolute values are always non-negative.\u201d Non-negative means greater than or equal to zero.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655164890\" data-type=\"problem\">\r\n<p id=\"fs-id1170655164892\">Simplify:\u00a0a)\u00a0\\(|3|\\)\u00a0b)\u00a0\\(|-44|\\)\u00a0c)\u00a0\\(|0|\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655213327\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<p id=\"fs-id1170654944089\">The absolute value of a number is the distance between the number and zero. Distance is never negative, so the absolute value is never negative.<\/p>\r\n<p id=\"fs-id1166424887501\">a) \\(|3|\\)\r\n\\(\\phantom{\\rule{1.2em}{0ex}}3\\)<\/p>\r\nb) \\(|-44|\\)\r\n\\(\\phantom{\\rule{1.5em}{0ex}}44\\)\r\n\r\nc) \\(|0|\\)\r\n\\(0\\phantom{\\rule{1.5em}{0ex}}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655074703\" data-type=\"problem\">\r\n<p id=\"fs-id1170655178086\">Simplify:\u00a0a)\u00a0\\(|4|\\)\u00a0b)\u00a0\\(|-28|\\)\u00a0c)\u00a0\\(|0|\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655161378\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170655161380\">a) 4 b) 28 c) 0<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654887750\">In the next example, we\u2019ll order expressions with absolute values. Remember, positive numbers are always greater than negative numbers!<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div data-type=\"problem\">\r\n<p id=\"fs-id1170654953584\">Fill in &lt;, &gt;, \\(\\phantom{\\rule{0.2em}{0ex}}\\text{or}\\phantom{\\rule{0.2em}{0ex}}=\\) for each of the following pairs of numbers:<\/p>\r\n<p id=\"fs-id1170654916476\">a) \\(|-5|\\)___\\(-|-5|\\) b) \\(8\\)___\\(-|-8|\\)c) \\(-9\\)___\\(-|-9|\\) d) -\\(-16\\)___\\(-|-16|\\)<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655106909\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\">\r\n<table style=\"border-collapse: collapse; width: 100%; height: 184px;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td><\/td>\r\n<td>\\(|-5|\\)___ -\\(|-5|\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 30px;\">\r\n<td>a) Simplify.\r\nOrder.<\/td>\r\n<td>5 ___ -5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td><\/td>\r\n<td>5\u00a0 &gt; -5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td><\/td>\r\n<td>\\(|-5|\\)\u00a0 &gt; -\\(|-5|\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td>b) Simplify.\r\nOrder.<\/td>\r\n<td>\\(8\\) ___ -\\(|-8|\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td><\/td>\r\n<td>8 ___ -8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td><\/td>\r\n<td>8 &gt; -8<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td><\/td>\r\n<td>8 &gt; -\\(|-8|\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td>c) Simplify.\r\nOrder.<\/td>\r\n<td>9 ___ -\\(|-9|\\)\r\n\r\n-9 ___ -9\r\n\r\n-9 = -9\r\n\r\n-9 = -\\(|-9|\\)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td>d) Simplify.\r\nOrder.<\/td>\r\n<td>-\\((-16)\\) ___ -\\(|-16|\\)\r\n\r\n16 ____ -16\r\n\r\n16 &gt; -16\r\n\r\n-\\((-16)\\) &gt; -\\(|-16|\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655121090\" data-type=\"problem\">\r\n<p id=\"fs-id1170655099637\">Fill in &lt;, &gt;, or \\(=\\) for each of the following pairs of numbers:\u00a0a) \\(|-9|\\)___-\\(|-9|\\) b) \\(2\\)___-\\(|-2|\\) c) \\(-8\\)___\\(|-8|\\)\r\nd) \\(\\-(-9\\)___\\(-|-9|\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655228722\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170655352028\">a) &gt; b) &gt; c) &lt; d) &gt;<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170655060167\">We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the <span class=\"no-emphasis\" data-type=\"term\">absolute value<\/span> of the resulting number.<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Grouping Symbols<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;\">Parentheses<\/td>\r\n<td style=\"width: 50%;\">( )<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Brackets<\/td>\r\n<td style=\"width: 50%;\">[ ]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Braces<\/td>\r\n<td style=\"width: 50%;\">{ }<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Absolute value<\/td>\r\n<td style=\"width: 50%;\">| |<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170655103794\">In the next example, we simplify the expressions inside absolute value bars first, just like we do with parentheses.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655218379\" data-type=\"problem\">\r\n<p id=\"fs-id1170655000668\">Simplify: \\(24-|19-3\\left(6-2\\right)|\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654968392\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<table id=\"eip-46\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(24-|19-3\\left(6-2\\right)|\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Work inside parentheses first: subtract 2 from 6.<\/td>\r\n<td>\\(24-|19-3\\left(4\\right)|\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply 3(4).<\/td>\r\n<td>\\(24-|19-12|\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract inside the absolute value bars.<\/td>\r\n<td>\\(24-|7|\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take the absolute value.<\/td>\r\n<td>\\(24-7\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>\\(17\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655222089\" data-type=\"problem\">\r\n<p id=\"fs-id1170655222091\">Simplify: \\(19-|11-4\\left(3-1\\right)|\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654963377\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654963379\">16<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Add Integers<\/h1>\r\n<p id=\"fs-id1170654942147\">Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.<\/p>\r\n<p id=\"fs-id1170654982815\">We will use two colour counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.<\/p>\r\n<p id=\"fs-id1170654940836\">We let one colour (blue) represent positive. The other colour (red) will represent the negatives. If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.<\/p>\r\n<span id=\"fs-id1170654940841\" data-type=\"media\" data-alt=\"In this image we have a blue counter above a red counter with a circle around both. The equation to the right is 1 plus negative 1 equals 0.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_014_img_new.jpg\" alt=\"In this image we have a blue counter above a red counter with a circle around both. The equation to the right is 1 plus negative 1 equals 0.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<p id=\"fs-id1170654925278\">We will use the counters to show how to add the four addition facts using the numbers \\(5,-5\\) and \\(3,-3\\).<\/p>\r\n\\(\\begin{array}{cccccccccc}\\hfill 5+3\\hfill &amp; &amp; &amp; \\hfill -5+\\left(-3\\right)\\hfill &amp; &amp; &amp; \\hfill -5+3\\hfill &amp; &amp; &amp; \\hfill 5+\\left(-3\\right)\\hfill \\end{array}\\)\r\n\r\nThe first example adds 5 positives and 3 positives\u2014both positives.\r\n\r\nThe second example adds 5 negatives and 3 negatives\u2014both negatives.\r\n<p id=\"fs-id1170654905046\">In each case we got 8\u2014either 8 positives or 8 negatives.<\/p>\r\n<p id=\"fs-id1170654905109\">When the signs were the same, the counters were all the same color, and so we added them.<\/p>\r\n<span id=\"fs-id1170655040356\" data-type=\"media\" data-alt=\"This figure is divided into two columns. In the left column there are eight blue counters in a horizontal row. Under them is the text \u201c8 positives.\u201d Centred under this is the equation 5 plus 3 equals 8. In the right column are eight red counters in a horizontal row which are labled below with the phrase \u201c8 negatives\u201d. Centred under this is the equation negative 5 plus negative 3 equals negative 8, where negative 3 is in parentheses.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_021_img_new.jpg\" alt=\"This figure is divided into two columns. In the left column there are eight blue counters in a horizontal row. Under them is the text \u201c8 positives.\u201d Centred under this is the equation 5 plus 3 equals 8. In the right column are eight red counters in a horizontal row which are labled below with the phrase \u201c8 negatives\u201d. Centred under this is the equation negative 5 plus negative 3 equals negative 8, where negative 3 is in parentheses.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<p id=\"fs-id1170655030242\">So what happens when the signs are different? Let\u2019s add \\(-5+3\\) and \\(5+\\left(-3\\right)\\).<\/p>\r\n<p id=\"fs-id1170655057773\">When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative.<\/p>\r\n<span id=\"fs-id1170655058518\" data-type=\"media\" data-alt=\"Two images are shown and labeled. The left image shows five red counters in a horizontal row drawn above three blue counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written \u201cnegative 5 plus 3\u201d and below is written \u201cMore negatives \u2013 the sum is negative.\u201d The right image shows five blue counters in a horizontal row drawn above three red counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written \u201c5 plus negative 3\u201d and below is written \u201cMore positives \u2013 the sum is positive.\u201d\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_026_img_new.jpg\" alt=\"Two images are shown and labeled. The left image shows five red counters in a horizontal row drawn above three blue counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written \u201cnegative 5 plus 3\u201d and below is written \u201cMore negatives \u2013 the sum is negative.\u201d The right image shows five blue counters in a horizontal row drawn above three red counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written \u201c5 plus negative 3\u201d and below is written \u201cMore positives \u2013 the sum is positive.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655058550\" data-type=\"problem\">\r\n<p id=\"fs-id1170655058552\">Add: a) \\(-1+5\\) b) \\(1+\\left(-5\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655029787\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<p id=\"fs-id1166422829183\">a)<\/p>\r\n\r\n<table id=\"eip-id1169754360753\" style=\"width: 100%;\" summary=\"In this image we have two columns. The top of the right column has the equation \u201cnegative 1 plus 5.\u201d The next line down in the left column reads, \u201cThere are more positives, so the sum is positive.\u201d Next to this in the right column are five blue counters. Under this, one red counter. The first blue counter and the single red counter are circled. Under this in the same column it says \u201c4.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"center\">\u22121 + 5<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><span id=\"eip-id1169749528166\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_027_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>There are more positives, so the sum is positive.<\/td>\r\n<td data-align=\"center\">4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nb)\r\n<table id=\"eip-id1169750608265\" style=\"width: 100%;\" summary=\"In this image we have two columns. The top of the right column has the equation \u201c1 plus negative 5.\u201d The next line down in the left column reads, \u201cThere are more negatives, so the sum is negative.\u201d Next to this in the right column is one blue counter. Under this, five red counters. The single blue counter and the first of the five red counters is circled. Under this in the same column it says \u201cnegative 4.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td data-align=\"center\">1 + (\u22125)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><span id=\"eip-id1169750608293\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_028_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>There are more negatives, so the sum is negative.<\/td>\r\n<td data-align=\"center\">\u22124<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654904893\" data-type=\"problem\">\r\n<p id=\"fs-id1170654904895\">Add:\u00a0a)\u00a0\\(-2+4\\)\u00a0b)\u00a0\\(2+\\left(-4\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655066666\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170655066668\">a) 2 b) \\(-2\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170655066171\">Now that we have added small positive and negative integers with a model, we can visualize the model in our minds to simplify problems with any numbers.<\/p>\r\n<p id=\"fs-id1170655067182\">When you need to add numbers such as \\(37+\\left(-53\\right)\\), you really don\u2019t want to have to count out 37 blue counters and 53 red counters. With the model in your mind, can you visualize what you would do to solve the problem?<\/p>\r\n<p id=\"fs-id1170655067203\">Picture 37 blue counters with 53 red counters lined up underneath. Since there would be more red (negative) counters than blue (positive) counters, the sum would be <em data-effect=\"italics\">negative<\/em>. How many more red counters would there be? Because \\(53-37=16\\), there are 16 more red counters.<\/p>\r\n<p id=\"fs-id1170655067279\">Therefore, the sum of \\(37+\\left(-53\\right)\\) is \\(-16\\).<\/p>\r\n\r\n<div id=\"fs-id1170655067436\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(37+\\left(-53\\right)=-16\\)<\/div>\r\n<p id=\"fs-id1170655069654\">Let\u2019s try another one. We\u2019ll add \\(-74+\\left(-27\\right)\\). Again, imagine 74 red counters and 27 more red counters, so we\u2019d have 101 red counters. This means the sum is \\(-101\\).<\/p>\r\n\\(-74+\\left(-27\\right)=-101\\)\r\n<p id=\"fs-id1170655069404\">Let\u2019s look again at the results of adding the different combinations of \\(5,-5\\) and \\(3,-3\\).<\/p>\r\n\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Addition of Positive and Negative Integers<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655069465\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{cccc}\\hfill 5+3\\hfill &amp; &amp; &amp; \\hfill -5+\\left(-3\\right)\\hfill \\\\ \\hfill 8\\hfill &amp; &amp; &amp; \\hfill -8\\hfill \\\\ \\hfill \\text{both positive, sum positive}\\hfill &amp; &amp; &amp; \\hfill \\text{both negative, sum negative}\\hfill \\end{array}\\)<\/div>\r\n<p id=\"fs-id1170655072721\">When the signs are the same, the counters would be all the same color, so add them.<\/p>\r\n\\(\\begin{array}{cccc}\\hfill -5+3\\hfill &amp; &amp; &amp; \\hfill 5+\\left(-3\\right)\\hfill \\\\ \\hfill -2\\hfill &amp; &amp; &amp; \\hfill 2\\hfill \\\\ \\hfill \\text{different signs, more negatives, sum negative}\\hfill &amp; &amp; &amp; \\hfill \\text{different signs, more positives, sum positive}\\hfill \\end{array}\\)\r\n<p id=\"fs-id1170655163407\">When the signs are different, some of the counters would make neutral pairs, so subtract to see how many are left.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654942464\">Visualize the model as you simplify the expressions in the following examples.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654932186\" data-type=\"problem\">\r\n<p id=\"fs-id1170654932188\">Simplify:\u00a0a)\u00a0\\(19+\\left(-47\\right)\\)\u00a0b)\u00a0\\(-14+\\left(-36\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654942539\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<ol id=\"fs-id1166422639606\" class=\"circled\" type=\"a\">\r\n \t<li>\u00a0Since the signs are different, we subtract \\(\\text{19 from 47}\\text{.}\\) The answer will be negative because there are more negatives than positives.\r\n\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{0.3em}{0ex}}19+\\left(-47\\right)\\hfill \\\\ \\text{Add.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{0.3em}{0ex}}-28\\hfill \\end{array}\\)<\/li>\r\n \t<li>\u00a0Since the signs are the same, we add. The answer will be negative because there are only negatives.\r\n\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill -14+\\left(-36\\right)\\hfill \\\\ \\text{Add.}\\hfill &amp; &amp; &amp; \\hfill -50\\hfill \\end{array}\\)<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654941250\" data-type=\"problem\">\r\n<p id=\"fs-id1170654941252\">Simplify:\u00a0a)\u00a0\\(-31+\\left(-19\\right)\\)\u00a0b)\u00a0\\(15+\\left(-32\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654941827\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170655215496\">a)\\(-50\\) b)\\(-17\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654944039\">The techniques used up to now extend to more complicated problems, like the ones we\u2019ve seen before. Remember to follow the order of operations!<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654929156\" data-type=\"problem\">\r\n<p id=\"fs-id1170654929159\">Simplify: \\(-5+3\\left(-2+7\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654934976\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<table id=\"eip-256\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(-5+3\\left(-2+7\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify inside the parentheses.<\/td>\r\n<td>\\(-5+3\\left(5\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>\\(-5+15\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add left to right.<\/td>\r\n<td>\\(10\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654936188\" data-type=\"problem\">\r\n<p id=\"fs-id1170654936190\">Simplify: \\(-2+5\\left(-4+7\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654944107\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654944110\">13<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Subtract Integers<\/h1>\r\n<p id=\"fs-id1170654921208\">We will continue to use counters to model the subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.<\/p>\r\n<p id=\"fs-id1170654921213\">Perhaps when you were younger, you read \\(\\text{\u201c}5-3\\text{\u201d}\\) as \\(\\text{\u201c}5\\) take away \\(3.\\text{\u201d}\\) When you use counters, you can think of subtraction the same way!<\/p>\r\n<p id=\"fs-id1170654932154\">We will model the four subtraction facts using the numbers \\(5\\) and \\(3\\).<\/p>\r\n\r\n<div id=\"fs-id1170654936672\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{cccccccccc}\\hfill 5-3\\hfill &amp; &amp; &amp; \\hfill -5-\\left(-3\\right)\\hfill &amp; &amp; &amp; \\hfill -5-3\\hfill &amp; &amp; &amp; \\hfill 5-\\left(-3\\right)\\hfill \\end{array}\\)<\/div>\r\nThe first example, we subtract 3 positives from 5 positives and end up with 2 positives.\r\n<p id=\"fs-id1170654942784\">In the second example, we subtract 3 negatives from 5 negatives and end up with 2 negatives.<\/p>\r\n<p id=\"fs-id1170654942787\">Each example used counters of only one color, and the \u201ctake away\u201d model of subtraction was easy to apply.<\/p>\r\n<span id=\"fs-id1170654942792\" data-type=\"media\" data-alt=\"Two images are shown and labeled. The first image shows five blue counters, three of which are circled with an arrow. Above the counters is the equation \u201c5 minus 3 equals 2.\u201d The second image shows five red counters, three of which are circled with an arrow. Above the counters is the equation \u201cnegative 5, minus, negative 3, equals negative 2.\u201d\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_031_img_new.jpg\" alt=\"Two images are shown and labeled. The first image shows five blue counters, three of which are circled with an arrow. Above the counters is the equation \u201c5 minus 3 equals 2.\u201d The second image shows five red counters, three of which are circled with an arrow. Above the counters is the equation \u201cnegative 5, minus, negative 3, equals negative 2.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<p id=\"fs-id1170654935830\">What happens when we have to subtract one positive and one negative number? We\u2019ll need to use both white and red counters as well as some <span class=\"no-emphasis\" data-type=\"term\">neutral pairs<\/span>. Adding a neutral pair does not change the value. It is like changing quarters to nickels\u2014the value is the same, but it looks different.<\/p>\r\n\r\n<ul id=\"fs-id1166424901740\" data-bullet-style=\"bullet\">\r\n \t<li>To subtract \\(-5-3\\), we restate it as \\(-5\\) take away 3.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1170654928179\">We start with 5 negatives. We need to take away 3 positives, but we do not have any positives to take away.<\/p>\r\n<p id=\"fs-id1170654928184\">Remember, a neutral pair has value zero. If we add 0 to 5 its value is still 5. We add neutral pairs to the 5 negatives until we get 3 positives to take away.<\/p>\r\n\r\n<table id=\"eip-id1169752192784\" style=\"width: 100%;\" summary=\"In this image, we have two columns and six rows. The first row in the right column reads, \u201cnegative 5 minus 3 means negative 5 take away 3.\u201d The next row down on the left reads, \u201cWe started with five negatives.\u201d Next to this are five red counters in a horizontal row with \u201cnegative\u201d five underneath. The next row reads, \u201cWe now add the neutrals needed to get three positives. Next to this in the right column are five red counters, a space, and three red counters. Under the three red counters are three blue counters. The next row down reads, \u201cWe remove the three positives. Next to this in the right column are the five red counters, a space, and three red counters. Under the three red counters are three blue counters with a circled arrow around them. The next row down reads, \u201cwe are left with eight negatives.\u201d Next to this in the right column in a horizontal row are five red counters, a space, followed by three red counters. Under the counters reads \u201c8 negatives\u201d. The last row in the left column reads, \u201cThe difference of negative 5 and 3 is negative 8.\u201d Next to this in the right column is the equation, \u201cnegative 5 minus 3equals negative eight.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\u22125 \u2212 3 means \u22125 take away 3.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We start with 5 negatives.<\/td>\r\n<td><span id=\"eip-id1169754016089\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_032a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We now add the neutrals needed to get 3 positives.<\/td>\r\n<td><span id=\"eip-id1169754016106\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_032b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We remove the 3 positives.<\/td>\r\n<td><span id=\"eip-id1169754016122\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_032c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We are left with 8 negatives.<\/td>\r\n<td><span id=\"eip-id1169754361424\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_032d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The difference of \u22125 and 3 is \u22128.<\/td>\r\n<td>\u22125 \u2212 3 = \u22128<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170655261587\">And now, the fourth case, \\(5-\\left(-3\\right)\\). We start with 5 positives. We need to take away 3 negatives, but there are no negatives to take away. So we add neutral pairs until we have 3 negatives to take away.<\/p>\r\n\r\n<table id=\"eip-id1169754361484\" style=\"width: 100%;\" summary=\"In this image, we have two columns and six rows. The first row in the right column reads, \u201c5 minus negative 3 means 5 take away negative 3.\u201d The next row down on the left reads, \u201cWe started with five positives.\u201d Next to this are five blue counters in a horizontal row. The next row reads, \u201cWe now add the needed neutral pairs. Next to this in the right column are five blue counters, a space, and three blue counters. Under the three blue counters are three red counters. The next row down reads, \u201cWe remove the 3 negatives. Next to this in the right column are the five blue counters, a space, and three blue counters. Under the three blue counters are three red counters with a circled arrow around them. The next row down reads, \u201cWe are left with eight positives.\u201d Next to this in the right column in a horizontal row are five blue counters, a space, followed by three blue counters. Under the counters reads \u201c8 positives\u201d. The last row in the left column reads, \u201cThe difference of 5 and negative 3 is 8.\u201d Next to this in the right column is the equation, \u201c5 minus negative 3equals eight.\u201d\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>5 \u2212 (\u22123) means 5 take away \u22123.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We start with 5 positives.<\/td>\r\n<td><span id=\"eip-id1169753968359\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_033a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We now add the needed neutrals pairs.<\/td>\r\n<td><span id=\"eip-id1169753968375\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_033b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We remove the 3 negatives.<\/td>\r\n<td><span id=\"eip-id1169753968392\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_033c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We are left with 8 positives.<\/td>\r\n<td><span id=\"eip-id1169752181606\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_033d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The difference of 5 and \u22123 is 8.<\/td>\r\n<td>5 \u2212 (\u22123) = 8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div data-type=\"example\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654941998\" data-type=\"problem\">\r\n<p id=\"fs-id1170654942000\">Subtract:\u00a0a)\u00a0\\(-3-1\\)\u00a0b)\u00a0\\(3-\\left(-1\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654943757\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<p id=\"fs-id1167829621734\">a)<\/p>\r\n\r\n<table id=\"eip-id1169749482384\" style=\"width: 100%;\" summary=\"Part \u201ca\u201d has three red counters, a space, and one red counter in a horizontal row. Next to this on the right is the equation \u201cnegative 3 minus 1.\u201d The next line down on the left reads,\u201dTake 1 positive from the one added neutral pair\u201d. Next to this on the right is a blue counter circled with an arrow. Next to this reads \u201cnegative 4.\u201d Part \u201cb\u201d has three blue counters, a space, and one blue counter in a horizontal row. Next to this on the right is the equation \u201c3 minus negative 1.\u201d The next line down on the left reads, \u201cTake one negative from the one added neutral pair.\u201d Next to this on the right is a red counter circled with an arrow. Next to this reads \u201c4\u201d.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Take 1 positive from the one added neutral pair.<\/td>\r\n<td><span id=\"eip-id1169749482408\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_034a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span>\r\n<span id=\"eip-id1169749482428\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_034b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<td data-valign=\"top\">\u22123 \u2212 1\r\n\r\n\u22124<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1167836289563\">b)<\/p>\r\n\r\n<table id=\"fs-id1167824767322\" style=\"width: 100%;\" summary=\"Part \u201cb\u201d has three blue counters, a space, and one blue counter in a horizontal row. Next to this on the right is the equation \u201c3 minus negative 1.\u201d The next line down on the left reads, \u201cTake one negative from the one added neutral pair.\u201d Next to this on the right is a red counter circled with an arrow. Next to this reads \u201c4\u201d.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Take 1 negative from the one added neutral pair.<\/td>\r\n<td><span id=\"eip-id1169749482451\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_034c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span>\r\n<span id=\"eip-id1169749482475\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_034d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\r\n<td data-align=\"center\" data-valign=\"top\">3 \u2212 (\u22121)\r\n\r\n4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654924800\" data-type=\"problem\">\r\n<p id=\"fs-id1170654941927\">Subtract:\u00a0a)\u00a0\\(-6-4\\)\u00a0b)\u00a0\\(6-\\left(-4\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654942432\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654942434\">a) \\(-10\\) b) 10<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\nHave you noticed that <em data-effect=\"italics\">subtraction of signed numbers can be done by adding the opposite<\/em>? In <a class=\"autogenerated-content\" href=\"#fs-id1170654941998\">Example 8<\/a>, \\(-3-1\\) is the same as \\(-3+\\left(-1\\right)\\) and \\(3-\\left(-1\\right)\\) is the same as \\(3+1\\). You will often see this idea, the subtraction property, written as follows:\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654903656\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Subtraction Property<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654903661\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(a-b=a+\\left(-b\\right)\\)<\/div>\r\n<div data-type=\"equation\" data-label=\"\"><\/div>\r\n<div class=\"unnumbered\" data-type=\"equation\" data-label=\"\">Subtracting a number is the same as adding its opposite.<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654903717\">Look at these two examples.<\/p>\r\n<span id=\"fs-id1170654962580\" data-type=\"media\" data-alt=\"Two images are shown and labeled. The first image shows four gray spheres drawn next to two gray spheres, where the four are circled in red, with a red arrow leading away to the lower left. This drawing is labeled above as \u201c6 minus 4\u201d and below as \u201c2.\u201d The second image shows four gray spheres and four red spheres, drawn one above the other and circled in red, with a red arrow leading away to the lower left, and two gray spheres drawn to the side of the four gray spheres. This drawing is labeled above as \u201c6 plus, open parenthesis, negative 4, close parenthesis\u201d and below as \u201c2.\u201d\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_035_img_new.jpg\" alt=\"Two images are shown and labeled. The first image shows four gray spheres drawn next to two gray spheres, where the four are circled in red, with a red arrow leading away to the lower left. This drawing is labeled above as \u201c6 minus 4\u201d and below as \u201c2.\u201d The second image shows four gray spheres and four red spheres, drawn one above the other and circled in red, with a red arrow leading away to the lower left, and two gray spheres drawn to the side of the four gray spheres. This drawing is labeled above as \u201c6 plus, open parenthesis, negative 4, close parenthesis\u201d and below as \u201c2.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<div id=\"fs-id1166424959329\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(6-4\\phantom{\\rule{0.2em}{0ex}}\\text{gives the same answer as}\\phantom{\\rule{0.2em}{0ex}}6+\\left(-4\\right)\\).<\/div>\r\n<p id=\"fs-id1170654962732\">Of course, when you have a subtraction problem that has only positive numbers, like \\(6-4\\), you just do the subtraction. You already knew how to subtract \\(6-4\\) long ago. But <em data-effect=\"italics\">knowing<\/em> that \\(6-4\\) gives the same answer as \\(6+\\left(-4\\right)\\) helps when you are subtracting negative numbers. Make sure that you understand how \\(6-4\\) and \\(6+\\left(-4\\right)\\) give the same results!<\/p>\r\n<p id=\"fs-id1170655075753\">Look at what happens when we subtract a negative.<\/p>\r\n<span id=\"fs-id1170655075756\" data-type=\"media\" data-alt=\"This figure is divided vertically into two halves. The left part of the figure contains the expression 8 minus negative 5, where negative 5 is in parentheses. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the group of five blue counters is a group of five red counters, which are circled. The circle has an arrow pointing away toward bottom left of the image, symbolizing subtraction. Below the counters is the number 13. The right part of the figure contains the expression 8 plus 5. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the counters is the number 13.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_036_img_new.jpg\" alt=\"This figure is divided vertically into two halves. The left part of the figure contains the expression 8 minus negative 5, where negative 5 is in parentheses. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the group of five blue counters is a group of five red counters, which are circled. The circle has an arrow pointing away toward bottom left of the image, symbolizing subtraction. Below the counters is the number 13. The right part of the figure contains the expression 8 plus 5. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the counters is the number 13.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<div id=\"fs-id1166424830896\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(8-\\left(-5\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{gives the same answer as}\\phantom{\\rule{0.2em}{0ex}}8+5\\)<\/div>\r\n<p id=\"fs-id1170655077076\"><em data-effect=\"italics\">Subtracting a negative number is like adding a positive!<\/em><\/p>\r\n<p id=\"fs-id1170655076379\">You will often see this written as \\(a-\\left(-b\\right)=a+b\\).<\/p>\r\n<p id=\"fs-id1170654962473\">What happens when there are more than three integers? We just use the order of operations as usual.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655084026\" data-type=\"problem\">\r\n<p id=\"fs-id1170655084028\">Simplify: \\(7-\\left(-4-3\\right)-9\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655083647\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<table id=\"eip-942\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(7-\\left(-4-3\\right)-9\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify inside the parentheses first.<\/td>\r\n<td>\\(7-\\left(-7\\right)-9\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract left to right.<\/td>\r\n<td>\\(14-9\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>\\(5\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170655091566\" data-type=\"problem\">\r\n<p id=\"fs-id1170655091568\">Simplify: \\(8-\\left(-3-1\\right)-9\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170655090951\" data-type=\"solution\"><details open=\"open\"><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170655090953\">3<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170655091187\" class=\"media-2\" data-type=\"note\">\r\n<h1>Multiply Integers<\/h1>\r\n<p id=\"fs-id1170654047593\">Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show <span class=\"no-emphasis\" data-type=\"term\">multiplication<\/span> of <span class=\"no-emphasis\" data-type=\"term\">integers<\/span>. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.<\/p>\r\n<p id=\"fs-id1170654027710\">We remember that \\(a \\cdot b\\) means add <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em> times. Here, we are using the model just to help us discover the pattern.<\/p>\r\n<span id=\"fs-id1170652629325\" data-type=\"media\" data-alt=\"Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads \u201cadd 5, 3 times.\u201d Below this depicts three rows of blue counters, with five counters in each row. Under this, it says \u201c15 positives.\u201d Under thisis the equation\u201c5 times 3 equals 15.\u201d The image on the right reads \u201cnegative 5 times three. The three is in parentheses. Below this it reads, \u201cadd negative five, three times.\u201d Under this are fifteen red counters in three rows of five. Below this it reads\u201d \u201c15 negatives\u201d. Below this is the equation negative five times 3 equals negative 15.\u201d\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_04_001_img_new.jpg\" alt=\"Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads \u201cadd 5, 3 times.\u201d Below this depicts three rows of blue counters, with five counters in each row. Under this, it says \u201c15 positives.\u201d Under thisis the equation\u201c5 times 3 equals 15.\u201d The image on the right reads \u201cnegative 5 times three. The three is in parentheses. Below this it reads, \u201cadd negative five, three times.\u201d Under this are fifteen red counters in three rows of five. Below this it reads\u201d \u201c15 negatives\u201d. Below this is the equation negative five times 3 equals negative 15.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<p id=\"fs-id1170654235984\">The next two examples are more interesting.<\/p>\r\n<p id=\"fs-id1171791114068\">What does it mean to multiply 5 by \\(-3?\\) It means subtract 5, 3 times. Looking at subtraction as \u201ctaking away,\u201d it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the work space. Then we take away 5 three times.<\/p>\r\n<span id=\"fs-id1170654116877\" data-type=\"media\" data-alt=\"This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.\"><img class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_04_002_img_new.jpg\" alt=\"This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n<p id=\"fs-id1170654009263\">In summary:<\/p>\r\n\r\n<div id=\"fs-id1170653916873\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\begin{array}{cccccccc}\\hfill 5 \\cdot 3&amp; =\\hfill &amp; 15\\hfill &amp; &amp; &amp; \\hfill -5\\left(3\\right)&amp; =\\hfill &amp; -15\\hfill \\\\ \\hfill 5\\left(-3\\right)&amp; =\\hfill &amp; -15\\hfill &amp; &amp; &amp; \\hfill \\left(-5\\right)\\left(-3\\right)&amp; =\\hfill &amp; 15\\hfill \\end{array}\\)<\/div>\r\n<p id=\"fs-id1170653885621\">Notice that for multiplication of two signed numbers, when the:<\/p>\r\n\r\n<ul id=\"fs-id1166425881422\" data-bullet-style=\"bullet\">\r\n \t<li>signs are the <em data-effect=\"italics\">same<\/em>, the product is <em data-effect=\"italics\">positive<\/em>.<\/li>\r\n \t<li>signs are <em data-effect=\"italics\">different<\/em>, the product is <em data-effect=\"italics\">negative<\/em>.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1170653867968\">We\u2019ll put this all together in the chart below<\/p>\r\n\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Multiplication of Signed Numbers<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor multiplication of two signed numbers:\r\n<table class=\"grid\" style=\"width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<th data-valign=\"top\" data-align=\"left\">Same signs<\/th>\r\n<td><strong>Product<\/strong><\/td>\r\n<td><strong>Example<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Two positives\r\nTwo negatives<\/td>\r\n<td>Positive\r\nPositive<\/td>\r\n<td>\\(\\begin{array}{ccc}\\hfill 7\\cdot4&amp; =\\hfill &amp; 28\\hfill \\\\ \\hfill -8\\left(-6\\right)&amp; =\\hfill &amp; 48\\hfill \\end{array}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table class=\"grid\" style=\"width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Different signs<\/strong><\/td>\r\n<td><strong>Product<\/strong><\/td>\r\n<td><strong>Example<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Positive \\cdot negative\r\nNegative \\cdot positive<\/td>\r\n<td>Negative\r\nNegative<\/td>\r\n<td>\\(\\begin{array}{ccc}\\hfill 7\\left(-9\\right)&amp; =\\hfill &amp; -63\\hfill \\\\ \\hfill -5\\cdot10&amp; =\\hfill &amp; -50\\hfill \\end{array}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654016481\" data-type=\"problem\">\r\n<p id=\"fs-id1170653756055\">Multiply: a) \\(-9\\cdot3\\) b) \\(-2\\left(-5\\right)\\) c) \\(4\\left(-8\\right)\\) d) \\(7\\cdot6\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654013170\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<table id=\"eip-959\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>a)\r\nMultiply, noting that the signs are different so the product is negative.<\/td>\r\n<td>\\(\\begin{array}{c}-9\\cdot3\\\\ -27\\end{array}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>b)\r\nMultiply, noting that the signs are the same so the product is positive.<\/td>\r\n<td>\\(\\begin{array}{c}-2\\left(-5\\right)\\\\ 10\\end{array}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>c)\r\nMultiply, with different signs.<\/td>\r\n<td>\\(\\begin{array}{c}4\\left(-8\\right)\\\\ -32\\end{array}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>d)\r\nMultiply, with same signs.<\/td>\r\n<td>\\(\\begin{array}{c}7\\cdot6\\\\ 42\\end{array}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654150864\" data-type=\"problem\">\r\n<p id=\"fs-id1170653870759\">Multiply: a) \\(-6\\cdot8\\) b) \\(-4\\left(-7\\right)\\) c) \\(9\\left(-7\\right)\\) d) \\(5\\cdot12\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654034104\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170653885313\">a)\\(-48\\) b) 28 c) \\(-63\\) d) 60<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170654150913\">When we multiply a number by 1, the result is the same number. What happens when we multiply a number by \\(-1\\)? Let\u2019s multiply a positive number and then a negative number by \\(-1\\) to see what we get.<\/p>\r\n\\(\\begin{array}{ccccccc}&amp; &amp; &amp; \\hfill -1\\cdot 4\\hfill &amp; &amp; &amp; \\hfill -1\\left(-3\\right)\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill -4\\hfill &amp; &amp; &amp; \\hfill 3\\hfill \\\\ &amp; &amp; &amp; \\hfill -4\\phantom{\\rule{0.2em}{0ex}}\\text{is the opposite of}\\phantom{\\rule{0.2em}{0ex}}4.\\hfill &amp; &amp; &amp; \\hfill 3\\phantom{\\rule{0.2em}{0ex}}\\text{is the opposite of}\\phantom{\\rule{0.2em}{0ex}}-3.\\hfill \\end{array}\\)\r\n<p id=\"fs-id1170652622975\">Each time we multiply a number by \\(-1\\), we get its opposite!<\/p>\r\n\r\n<div id=\"fs-id1170653776943\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Multiplication by \\(-1\\)<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n\\(-1a=\\text{-}a\\)\r\n\r\nMultiplying a number by \\(-1\\) gives its opposite.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654068942\" data-type=\"problem\">\r\n<p id=\"fs-id1170654036219\">Multiply: a) \\(-1\\cdot7\\) b) \\(-1\\left(-11\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653910210\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<table id=\"eip-228\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>a)\r\nMultiply, noting that the signs are different so the product is negative.<\/td>\r\n<td>\\(\\begin{array}{c}-1\\cdot7\\\\ -7\\\\ -7\\phantom{\\rule{0.2em}{0ex}}\\text{is the opposite of}\\phantom{\\rule{0.2em}{0ex}}7.\\end{array}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>b)\r\nMultiply, noting that the signs are the same so the product is positive.<\/td>\r\n<td>\\(\\begin{array}{c}-1\\left(-11\\right)\\\\ 11\\\\ 11\\phantom{\\rule{0.2em}{0ex}}\\text{is the opposite of}\\phantom{\\rule{0.2em}{0ex}}-11.\\end{array}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654073784\" data-type=\"problem\">\r\n<p id=\"fs-id1170653851085\">Multiply: a) \\(-1\\cdot9\\) b) \\(-1 \\cdot \\left(-17\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654014381\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654231240\">a)\\(-9\\) b) 17<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Divide Integers<\/h1>\r\n<p id=\"fs-id1170654029961\">What about <span class=\"no-emphasis\" data-type=\"term\">division<\/span>? Division is the inverse operation of multiplication. So, \\(15\\div 3=5\\) because \\(5\\cdot3=15\\). In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.<\/p>\r\n\\(\\begin{array}{cccccccccccccc}\\hfill 5\\cdot3&amp; =\\hfill &amp; 15\\phantom{\\rule{0.2em}{0ex}}\\text{so}\\phantom{\\rule{0.2em}{0ex}}15\\div 3\\hfill &amp; =\\hfill &amp; 5\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill -5\\left(3\\right)&amp; =\\hfill &amp; -15\\phantom{\\rule{0.2em}{0ex}}\\text{so}\\phantom{\\rule{0.2em}{0ex}}-15\\div3\\hfill &amp; =\\hfill &amp; -5\\hfill \\\\ \\hfill \\left(-5\\right)\\left(-3\\right)&amp; =\\hfill &amp; 15\\phantom{\\rule{0.2em}{0ex}}\\text{so}\\phantom{\\rule{0.2em}{0ex}}15\\div \\left(-3\\right)\\hfill &amp; =\\hfill &amp; -5\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill 5\\left(-3\\right)&amp; =\\hfill &amp; -15\\phantom{\\rule{0.2em}{0ex}}\\text{so}\\phantom{\\rule{0.2em}{0ex}}-15\\div \\left(-3\\right)\\hfill &amp; =\\hfill &amp; 5\\hfill \\end{array}\\)\r\n<p id=\"fs-id1170654116562\">Division follows the same rules as multiplication!<\/p>\r\n<p id=\"fs-id1170653915002\">For division of two signed numbers, when the:<\/p>\r\n\r\n<ul id=\"fs-id1166426287557\" data-bullet-style=\"bullet\">\r\n \t<li>signs are the <em data-effect=\"italics\">same<\/em>, the quotient is <em data-effect=\"italics\">positive<\/em>.<\/li>\r\n \t<li>signs are <em data-effect=\"italics\">different<\/em>, the quotient is <em data-effect=\"italics\">negative<\/em>.<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1170653869100\">And remember that we can always check the answer of a division problem by multiplying.<\/p>\r\n\r\n<div id=\"fs-id1170654015283\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Multiplication and Division of Signed Numbers<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p id=\"fs-id1170653774421\">For multiplication and division of two signed numbers:<\/p>\r\n\r\n<ul id=\"fs-id1166425927232\" data-bullet-style=\"bullet\">\r\n \t<li>If the signs are the same, the result is positive.<\/li>\r\n \t<li>If the signs are different, the result is negative.<\/li>\r\n<\/ul>\r\n<table id=\"fs-id1170654154997\" style=\"width: 100%;\" summary=\"No Summary\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th style=\"width: 50%;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Same signs<\/th>\r\n<th style=\"width: 50%;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Result<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td style=\"width: 50%;\" data-valign=\"top\" data-align=\"left\">Two positives<\/td>\r\n<td style=\"width: 50%;\" data-valign=\"top\" data-align=\"left\">Positive<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\">Two negatives<\/td>\r\n<td style=\"width: 50%;\">Positive<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIf the signs are the same, the result is positive.\r\n<table id=\"fs-id1170653799921\" style=\"width: 100%; height: 48px;\" summary=\"No Summary\">\r\n<thead>\r\n<tr style=\"height: 16px;\" valign=\"top\">\r\n<th style=\"height: 16px; width: 50%;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Different signs<\/th>\r\n<th style=\"height: 16px; width: 50%;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Result<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 16px;\" valign=\"top\">\r\n<td style=\"height: 16px; width: 50%;\" data-valign=\"top\" data-align=\"left\">Positive and negative<\/td>\r\n<td style=\"height: 16px; width: 50%;\" data-valign=\"top\" data-align=\"left\">Negative<\/td>\r\n<\/tr>\r\n<tr style=\"height: 16px;\">\r\n<td style=\"height: 16px; width: 50%;\">Negative and positive<\/td>\r\n<td style=\"height: 16px; width: 50%;\">Negative<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIf the signs are different, the result is negative.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 12<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170652618826\" data-type=\"problem\">\r\n<p id=\"fs-id1170654008734\">Divide: a) \\(-27\\div 3\\) b) \\(-100\\div \\left(-4\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654212229\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<table id=\"eip-762\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>a)\r\nDivide. With different signs, the quotient is negative.<\/td>\r\n<td>\\(\\begin{array}{c}-27\\div 3\\\\ -9\\end{array}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>b)\r\nDivide. With signs that are the same, the quotient is positive.<\/td>\r\n<td>\\(\\begin{array}{c}-100\\div\\left(-4\\right)\\\\ 25\\end{array}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 12<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170653797425\" data-type=\"problem\">\r\n<p id=\"fs-id1170653906270\">Divide: a) \\(-42\\div 6\\) b) \\(-117\\div\\left(-3\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654149901\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654047662\">a)\\(-7\\) b) 39<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h1>Simplify Expressions with Integers<\/h1>\r\n<p id=\"fs-id1170653741550\">What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?<\/p>\r\n<p id=\"fs-id1170653790201\">Let\u2019s try some examples. We\u2019ll simplify expressions that use all four operations with integers\u2014addition, subtraction, multiplication, and division. Remember to follow the order of operations.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 13<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654066443\" data-type=\"problem\">\r\n<p id=\"fs-id1170653768124\">Simplify: \\(7\\left(-2\\right)+4\\left(-7\\right)-6\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170652623765\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<table id=\"eip-431\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(7\\left(-2\\right)+4\\left(-7\\right)-6\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply first.<\/td>\r\n<td>\\(-14+\\left(-28\\right)-6\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>\\(-42-6\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>\\(-48\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 13<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170652620764\" data-type=\"problem\">\r\n<p id=\"fs-id1170653875298\">Simplify: \\(8\\left(-3\\right)+5\\left(-7\\right)-4\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654149857\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654149859\">\\(-63\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170653871378\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1170652630281\" data-type=\"exercise\">\r\n<div id=\"fs-id1170652620232\" data-type=\"solution\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 14<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170652619970\" data-type=\"problem\">\r\n<p id=\"fs-id1170653761875\">Simplify: a) \\({\\left(-2\\right)}^{4}\\) b) \\(\\text{-}{2}^{4}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654237484\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<table id=\"eip-217\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>a)\r\nWrite in expanded form.\r\nMultiply.\r\nMultiply.\r\nMultiply.<\/td>\r\n<td>\\(\\begin{array}{c}{\\left(-2\\right)}^{4}\\\\ \\left(-2\\right)\\left(-2\\right)\\left(-2\\right)\\left(-2\\right)\\\\ 4\\left(-2\\right)\\left(-2\\right)\\\\ -8\\left(-2\\right)\\\\ 16\\end{array}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>b)\r\nWrite in expanded form. We are asked to find the opposite of\\(\\phantom{\\rule{0.2em}{0ex}}{2}^{4}\\).\r\nMultiply.\r\nMultiply.\r\nMultiply.<\/td>\r\n<td>\\(\\begin{array}{c}\\text{-}{2}^{4}\\\\ \\text{-}\\left(2\\cdot2\\cdot2\\cdot2\\right)\\\\ \\text{-}\\left(4\\cdot2\\cdot2\\right)\\\\ \\text{-}\\left(8\\cdot2\\right)\\\\ 16\\end{array}\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1170653737864\">Notice the difference in parts a) and b). In part a), the exponent means to raise what is in the parentheses, the \\(\\left(-2\\right)\\) to the \\({4}^{\\text{th}}\\) power. In part b), the exponent means to raise just the 2 to the \\({4}^{\\text{th}}\\) power and then take the opposite.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 14<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170652622523\" data-type=\"problem\">\r\n<p id=\"fs-id1170653790279\">Simplify: a) \\({\\left(-3\\right)}^{4}\\) b) \\(\\text{-}{3}^{4}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654230619\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654230621\">a) 81 b) \\(-81\\)<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170653904816\">The next example reminds us to simplify inside parentheses first.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 15<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170652622789\" data-type=\"problem\">\r\n<p id=\"fs-id1170652622791\">Simplify: \\(12-3\\left(9-12\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653905533\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<table id=\"eip-138\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(12-3\\left(9-12\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract in parentheses first.<\/td>\r\n<td>\\(12-3\\left(-3\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>\\(12-\\left(-9\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>\\(21\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 15<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654134940\" data-type=\"problem\">\r\n<p id=\"fs-id1170654134942\">Simplify: \\(17-4\\left(8-11\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170652622244\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654150070\">29<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 16<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170652624014\" data-type=\"problem\">\r\n<p id=\"fs-id1170652624016\">Simplify: \\(8\\left(-9\\right)\\div{\\left(-2\\right)}^{3}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653850514\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<div data-type=\"title\"><\/div>\r\n<table id=\"eip-410\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(8\\left(-9\\right)\\div{\\left(-2\\right)}^{3}\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Exponents first.<\/td>\r\n<td>\\(8\\left(-9\\right)\\div\\left(-8\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>\\(-72\\div \\left(-8\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>\\(9\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 16<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170652621179\" data-type=\"problem\">\r\n<p id=\"fs-id1170652621181\">Simplify: \\(12\\left(-9\\right)\\div{\\left(-3\\right)}^{3}\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170653797293\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170653797295\">4<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1170654156593\" class=\"try\" data-type=\"note\">\r\n<div id=\"fs-id1170652621176\" data-type=\"exercise\">\r\n<div id=\"fs-id1170652621179\" data-type=\"problem\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">EXAMPLE 17<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654047271\" data-type=\"problem\">\r\n<p id=\"fs-id1170654047273\">Simplify: \\(-30\\div 2+\\left(-3\\right)\\left(-7\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654233306\" data-type=\"solution\">\r\n<div data-type=\"title\"><\/div>\r\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\r\n<table id=\"eip-343\" style=\"width: 100%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\\(-30\\div 2+\\left(-3\\right)\\left(-7\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply and divide left to right, so divide first.<\/td>\r\n<td>\\(-15+\\left(-3\\right)\\left(-7\\right)\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>\\(-15+21\\)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>\\(6\\)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">TRY IT 17<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div id=\"fs-id1170654154229\" data-type=\"problem\">\r\n<p id=\"fs-id1170654154231\">Simplify: \\(-27\\div 3+\\left(-5\\right)\\left(-6\\right)\\).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1170654079166\" data-type=\"solution\"><details><summary>Show answer<\/summary>\r\n<p id=\"fs-id1170654079169\">21<\/p>\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170655091191\">Access these online resources for additional instruction and practice with adding and subtracting integers. You will need to enable Java in your web browser to use the applications.<\/p>\r\n\r\n<ul id=\"fs-id1170655096461\" data-display=\"block\">\r\n \t<li><a href=\"https:\/\/openstax.org\/l\/11AddColorChip\">Add Colored Chip<\/a><\/li>\r\n \t<li><a href=\"https:\/\/openstax.org\/l\/11SubtrColorChp\">Subtract Colored Chip<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<h1>Key Concepts<\/h1>\r\n<ul id=\"fs-id1170655097500\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Addition of Positive and Negative Integers<\/strong>\r\n\\(\\begin{array}{cccc}5+3\\hfill &amp; &amp; &amp; \\phantom{\\rule{2.5em}{0ex}}-5+\\left(-3\\right)\\hfill \\\\ \\phantom{\\rule{0.95em}{0ex}}8\\hfill &amp; &amp; &amp; \\phantom{\\rule{3.45em}{0ex}}-8\\hfill \\\\ \\text{both positive,}\\hfill &amp; &amp; &amp; \\phantom{\\rule{2.5em}{0ex}}\\text{both negative,}\\hfill \\\\ \\text{sum positive}\\hfill &amp; &amp; &amp; \\phantom{\\rule{2.5em}{0ex}}\\text{sum negative}\\hfill \\\\ \\\\ \\\\ -5+3\\hfill &amp; &amp; &amp; \\phantom{\\rule{2.5em}{0ex}}5+\\left(-3\\right)\\hfill \\\\ \\phantom{\\rule{0.95em}{0ex}}-2\\hfill &amp; &amp; &amp; \\phantom{\\rule{3.45em}{0ex}}2\\hfill \\\\ \\text{different signs,}\\hfill &amp; &amp; &amp; \\phantom{\\rule{2.5em}{0ex}}\\text{different signs,}\\hfill \\\\ \\text{more negatives}\\hfill &amp; &amp; &amp; \\phantom{\\rule{2.5em}{0ex}}\\text{more positives}\\hfill \\\\ \\text{sum negative}\\hfill &amp; &amp; &amp; \\phantom{\\rule{2.5em}{0ex}}\\text{sum positive}\\hfill \\end{array}\\)<\/li>\r\n \t<li><strong data-effect=\"bold\">Property of Absolute Value<\/strong>: \\(|n|\\ge 0\\) for all numbers. Absolute values are always greater than or equal to zero!<\/li>\r\n \t<li><strong data-effect=\"bold\">Subtraction of Integers<\/strong>\r\n\\(\\begin{array}{cccc}5-3\\hfill &amp; &amp; &amp; -5-\\left(-3\\right)\\hfill \\\\ \\phantom{\\rule{0.95em}{0ex}}2\\hfill &amp; &amp; &amp; \\phantom{\\rule{0.95em}{0ex}}-2\\hfill \\\\ 5\\phantom{\\rule{0.2em}{0ex}}\\text{positives}\\hfill &amp; &amp; &amp; 5\\phantom{\\rule{0.2em}{0ex}}\\text{negatives}\\hfill \\\\ \\text{take away}\\phantom{\\rule{0.2em}{0ex}}3\\phantom{\\rule{0.2em}{0ex}}\\text{positives}\\hfill &amp; &amp; &amp; \\text{take away}\\phantom{\\rule{0.2em}{0ex}}3\\phantom{\\rule{0.2em}{0ex}}\\text{negatives}\\hfill \\\\ \\text{2 positives}\\hfill &amp; &amp; &amp; \\text{2 negatives}\\hfill \\\\ \\\\ \\\\ -5-3\\hfill &amp; &amp; &amp; 5-\\left(-3\\right)\\hfill \\\\ \\phantom{\\rule{0.95em}{0ex}}-8\\hfill &amp; &amp; &amp; \\phantom{\\rule{0.95em}{0ex}}8\\hfill \\\\ 5\\phantom{\\rule{0.2em}{0ex}}\\text{negatives, want to}\\hfill &amp; &amp; &amp; 5\\phantom{\\rule{0.2em}{0ex}}\\text{positives, want to}\\hfill \\\\ \\text{subtract}\\phantom{\\rule{0.2em}{0ex}}3\\phantom{\\rule{0.2em}{0ex}}\\text{positives}\\hfill &amp; &amp; &amp; \\text{subtract}\\phantom{\\rule{0.2em}{0ex}}3\\phantom{\\rule{0.2em}{0ex}}\\text{negatives}\\hfill \\\\ \\text{need neutral pairs}\\hfill &amp; &amp; &amp; \\text{need neutral pairs}\\hfill \\end{array}\\)<\/li>\r\n \t<li><strong data-effect=\"bold\">Subtraction Property:<\/strong> Subtracting a number is the same as adding its opposite.<\/li>\r\n \t<li><strong data-effect=\"bold\">Multiplication and Division of Two Signed Numbers<\/strong>\r\n<ul id=\"fs-id1170653881237\" data-bullet-style=\"open-circle\">\r\n \t<li>Same signs\u2014Product is positive<\/li>\r\n \t<li>Different signs\u2014Product is negative<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<h1>Glossary<\/h1>\r\n<div class=\"textbox shaded\">\r\n<dl id=\"fs-id1166424847478\">\r\n \t<dt>absolute value<\/dt>\r\n \t<dd id=\"fs-id1166424809754\">The absolute value of a number is its distance from 0 on the number line. The absolute value of a number \\(n\\) is written as \\(|n|\\).<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1166424772590\">\r\n \t<dt>integers<\/dt>\r\n \t<dd id=\"fs-id1166424772595\">The whole numbers and their opposites are called the integers: ...\u22123, \u22122, \u22121, 0, 1, 2, 3...<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1166422694257\">\r\n \t<dt>opposite<\/dt>\r\n \t<dd id=\"fs-id1166422694262\">The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero: \\(-a\\) means the opposite of the number. The notation \\(-a\\) is read \u201cthe opposite of \\(a\\).\u201d<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<h1>1.2 Exercise Set<\/h1>\r\n<p id=\"fs-id1166422643633\">In the following exercises, order each of the following pairs of numbers, using &lt; or &gt;.<\/p>\r\n\r\n<ol>\r\n \t<li>\r\n<ol type=\"a\">\r\n \t<li>\\(9\\)___\\(4\\)<\/li>\r\n \t<li>\\(-3\\)___\\(6\\)<\/li>\r\n \t<li>\\(-8\\)___\\(-2\\)<\/li>\r\n \t<li>\\(1\\)___\\(-10\\)<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1171791388513\">In the following exercises, simplify.<\/p>\r\n\r\n<ol start=\"2\">\r\n \t<li>\r\n<ol type=\"a\">\r\n \t<li>\\(|-32|\\)<\/li>\r\n \t<li>\\(|0|\\)<\/li>\r\n \t<li>\\(|16|\\)<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170655355498\">In the following exercises, fill in &lt;, &gt;, or \\(=\\) for each of the following pairs of numbers.<\/p>\r\n\r\n<ol start=\"3\">\r\n \t<li>\r\n<ol type=\"a\">\r\n \t<li>\\(-6\\)___\\(|-6|\\)<\/li>\r\n \t<li>\\(-|-3|\\)___\\(-3\\)<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170655215530\">In the following exercises, simplify.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"4\">\r\n \t<li>\\(-\\left(-5\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}-|-5|\\)<\/li>\r\n \t<li>\\(8|-7|\\)<\/li>\r\n \t<li>\\(|15-7|-|14-6|\\)<\/li>\r\n \t<li>\\(18-|2\\left(8-3\\right)|\\)<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1166424899765\">In the following exercises, simplify each expression.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"8\">\r\n \t<li>\\(-21+\\left(-59\\right)\\)<\/li>\r\n \t<li>\\(48+\\left(-16\\right)\\)<\/li>\r\n \t<li>\\(-14+\\left(-12\\right)+4\\)<\/li>\r\n \t<li>\\(135+\\left(-110\\right)+83\\)<\/li>\r\n \t<li>\\(19+2\\left(-3+8\\right)\\)<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1166422656200\">In the following exercises, simplify.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"13\">\r\n \t<li>\\(8-2\\)<\/li>\r\n \t<li>\\(-5-4\\)<\/li>\r\n \t<li>\\(8-\\left(-4\\right)\\)<\/li>\r\n \t<li>\r\n<ol type=\"a\">\r\n \t<li>\\(44-28\\)<\/li>\r\n \t<li>\\(44+\\left(-28\\right)\\)<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>\r\n<ol type=\"a\">\r\n \t<li>\\(27-\\left(-18\\right)\\)<\/li>\r\n \t<li>\\(27+18\\)<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1170655166333\">In the following exercises, simplify each expression.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"18\">\r\n \t<li>\\(15-\\left(-12\\right)\\)<\/li>\r\n \t<li>\\(48-87\\)<\/li>\r\n \t<li>\\(-17-42\\)<\/li>\r\n \t<li>\\(-103-\\left(-52\\right)\\)<\/li>\r\n \t<li>\\(-45-\\left(54\\right)\\)<\/li>\r\n \t<li>\\(8-3-7\\)<\/li>\r\n \t<li>\\(-5-4+7\\)<\/li>\r\n \t<li>\\(-14-\\left(-27\\right)+9\\)<\/li>\r\n \t<li>\\(\\left(2-7\\right)-\\left(3-8\\right)\\left(2\\right)\\)<\/li>\r\n \t<li>\\(-\\left(6-8\\right)-\\left(2-4\\right)\\)<\/li>\r\n \t<li>\\(25-\\left[10-\\left(3-12\\right)\\right]\\)<\/li>\r\n \t<li>\\(6.3-4.3-7.2\\)<\/li>\r\n \t<li>\\({5}^{2}-{6}^{2}\\)<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1166421558397\">In the following exercises, multiply.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"31\">\r\n \t<li>\\(-4\\cdot8\\)<\/li>\r\n \t<li>\\(-1\\cdot6\\)<\/li>\r\n \t<li>\\(9\\left(-7\\right)\\)<\/li>\r\n \t<li>\\(-1\\left(-14\\right)\\)<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1166421949046\">In the following exercises, divide.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"35\">\r\n \t<li>\\(-24\\div6\\)<\/li>\r\n \t<li>\\(-180\\div15\\)<\/li>\r\n \t<li>\\(-52\\div\\left(-4\\right)\\)<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1166421486016\">In the following exercises, simplify each expression.<\/p>\r\n\r\n<ol class=\"twocolumn\" start=\"38\">\r\n \t<li>\\(5\\left(-6\\right)+7\\left(-2\\right)-3\\)<\/li>\r\n \t<li>\\({\\left(-2\\right)}^{6}\\)<\/li>\r\n \t<li>\\(\\text{-}{4}^{2}\\)<\/li>\r\n \t<li>\\(-3\\left(-5\\right)\\left(6\\right)\\)<\/li>\r\n \t<li>\\(\\left(8-11\\right)\\left(9-12\\right)\\)<\/li>\r\n \t<li>\\(26-3\\left(2-7\\right)\\)<\/li>\r\n \t<li>\\(65\\div\\left(-5\\right)+\\left(-28\\right)\\div\\left(-7\\right)\\)<\/li>\r\n \t<li>\\(9-2\\left[3-8\\left(-2\\right)\\right]\\)<\/li>\r\n \t<li>\\({\\left(-3\\right)}^{2}-24\\div\\left(8-2\\right)\\)<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1166423907674\">In the following exercises, solve.<\/p>\r\n\r\n<ol start=\"47\">\r\n \t<li><strong data-effect=\"bold\">Temperature<\/strong> On January \\(15\\), the high temperature in Lytton, British Columbia, was \\(84\\)\u00b0 . That same day, the high temperature in Fort Nelson, British Columbia was \\(-12\\)\u00b0. What was the difference between the temperature in Lytton and the temperature in Embarrass?<\/li>\r\n \t<li><strong data-effect=\"bold\"> Checking Account<\/strong> Ester has \\$124 in her checking account. She writes a check for \\$152. What is the new balance in her checking account?<\/li>\r\n \t<li><strong data-effect=\"bold\">Checking Account<\/strong> Kevin has a balance of \\(-\\$38\\) in his checking account. He deposits \\$225 to the account. What is the new balance?<\/li>\r\n \t<li>\r\n<p id=\"fs-id1170654920103\"><strong>Provincial budgets<\/strong> For 2019 the province of Quebec estimated it would have a budget surplus of \\$5.6 million. That same year, Alberta estimated it would have a budget deficit of \\$7.5 million.<\/p>\r\n<p id=\"fs-id1170655202258\">Use integers to write the budget of:<\/p>\r\n\r\n<ol type=\"a\">\r\n \t<li>Quebec<\/li>\r\n \t<li>Alberta<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<h1>Answers:<\/h1>\r\n<ol class=\"threecolumn\">\r\n \t<li>\r\n<ol type=\"a\">\r\n \t<li>&gt;<\/li>\r\n \t<li>&lt;<\/li>\r\n \t<li>&lt;<\/li>\r\n \t<li>&gt;<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>\r\n<ol type=\"a\">\r\n \t<li>32<\/li>\r\n \t<li>0<\/li>\r\n \t<li>16<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>\r\n<ol type=\"a\">\r\n \t<li>&lt;<\/li>\r\n \t<li>=<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>\\(5,-5\\)<\/li>\r\n \t<li>56<\/li>\r\n \t<li>0<\/li>\r\n \t<li>8<\/li>\r\n \t<li>\\(-80\\)<\/li>\r\n \t<li>32<\/li>\r\n \t<li>\\(-22\\)<\/li>\r\n \t<li>108<\/li>\r\n \t<li>29<\/li>\r\n \t<li>6<\/li>\r\n \t<li>\\(-9\\)<\/li>\r\n \t<li>12<\/li>\r\n \t<li>\r\n<ol type=\"a\">\r\n \t<li>16<\/li>\r\n \t<li>16<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>\r\n<ol type=\"a\">\r\n \t<li>45<\/li>\r\n \t<li>45<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>27<\/li>\r\n \t<li>\\(-39\\)<\/li>\r\n \t<li>\\(-59\\)<\/li>\r\n \t<li>\\(-51\\)<\/li>\r\n \t<li>-99<\/li>\r\n \t<li>\\(-2\\)<\/li>\r\n \t<li>\\(-2\\)<\/li>\r\n \t<li>22<\/li>\r\n \t<li>\\(-15\\)<\/li>\r\n \t<li>4<\/li>\r\n \t<li>6<\/li>\r\n \t<li>\\(-5.2\\)<\/li>\r\n \t<li>\\(-11\\)<\/li>\r\n \t<li>\\(-32\\)<\/li>\r\n \t<li>\\(-6\\)<\/li>\r\n \t<li>\\(-63\\)<\/li>\r\n \t<li>14<\/li>\r\n \t<li>\\(-4\\)<\/li>\r\n \t<li>\\(-12\\)<\/li>\r\n \t<li>13<\/li>\r\n \t<li>\\(-47\\)<\/li>\r\n \t<li>64<\/li>\r\n \t<li>\\(-16\\)<\/li>\r\n \t<li>90<\/li>\r\n \t<li>9<\/li>\r\n \t<li>41<\/li>\r\n \t<li>\\(-9\\)<\/li>\r\n \t<li>\\(-29\\)<\/li>\r\n \t<li>5<\/li>\r\n \t<li>\\(96\\)\u00b0<\/li>\r\n \t<li>\\(-\\$28\\)<\/li>\r\n \t<li>\\$187<\/li>\r\n \t<li>\r\n<ol type=\"a\">\r\n \t<li>\\$5.6 million<\/li>\r\n \t<li>\\(-\\$7.5\\) million<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<div data-type=\"glossary\">\r\n<h1>Attributions<\/h1>\r\nThis chapter has been adapted from \u201cAdd and Subtract Integers\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em><\/a> (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. See the Copyright page for more information.\r\n\r\n<\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section it is expected that you will be able to:<\/p>\n<ul>\n<li>Use negatives and opposites<\/li>\n<li>Simplify: expressions with absolute value<\/li>\n<li>Add ans subtract integers<\/li>\n<li>Multiply and divide integers<\/li>\n<li>Simplify Expressions with Integers<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1>Use Negatives and Opposites<\/h1>\n<p id=\"fs-id1170655222181\">If you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. <strong data-effect=\"bold\">Negative numbers<\/strong> are numbers less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>. The <span class=\"no-emphasis\" data-type=\"term\">negative numbers<\/span> are to the left of zero on the number line. See <a class=\"autogenerated-content\" href=\"#figure1\">Figure 1<\/a>.<a id=\"figure1\"><\/a><\/p>\n<figure style=\"width: 389px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_001_new.jpg\" alt=\"A number line extends from negative 4 to 4. A bracket is under the values \u201cnegative 4\u201d to \u201c0\u201d and is labeled \u201cNegative numbers\u201d. Another bracket is under the values 0 to 4 and labeled \u201cpositive numbers\u201d. There is an arrow in between both brackets pointing upward to zero.\" width=\"389\" height=\"113\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 1 The number line shows the location of positive and negative numbers.<\/figcaption><\/figure>\n<p>The arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number, and there is no smallest negative number.<\/p>\n<p id=\"fs-id1170655007025\">Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative.<\/p>\n<p>Consider how numbers are ordered on the number line. Going from left to right, the numbers increase in value. Going from right to left, the numbers decrease in value. See <a class=\"autogenerated-content\" href=\"#figure2\">Figure 2<\/a>.<a id=\"figure2\"><\/a><\/p>\n<div id=\"CNX_ElemAlg_Figure_01_03_002_new\" class=\"bc-figure figure\">\n<figure style=\"width: 440px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_002_new.jpg\" alt=\"A number line ranges from negative 4 to 4. An arrow above the number line extends from negative 1 towards 4 and is labeled \u201clarger\u201d. An arrow below the number line extends from 1 towards negative 4 and is labeled \u201csmaller\u201d.\" width=\"440\" height=\"102\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 2 The numbers on a number line increase in value going from left to right and decrease in value going from right to left.<\/figcaption><\/figure>\n<\/div>\n<div class=\"textbox shaded\">\n<p>Remember that we use the notation:<\/p>\n<p id=\"fs-id1170654965355\"><em data-effect=\"italics\">a<\/em> &lt; <em data-effect=\"italics\">b<\/em> (read \u201ca is less than b\u201d) when a is to the left of b on the number line.<\/p>\n<p id=\"fs-id1170654901053\"><em data-effect=\"italics\">a<\/em> &gt; <em data-effect=\"italics\">b<\/em> (read \u201c<em data-effect=\"italics\">a<\/em> is greater than <em data-effect=\"italics\">b<\/em>\u201d) when <em data-effect=\"italics\">a<\/em> is to the right of <em data-effect=\"italics\">b<\/em> on the number line.<\/p>\n<\/div>\n<p id=\"fs-id1170654965087\">Now we need to extend the number line which showed the whole numbers to include negative numbers, too. The numbers marked by points in <a class=\"autogenerated-content\" href=\"#figure3\">Figure 3<\/a> are called the integers. The integers are the numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-759e65d987e3039dac89a95f0411c0c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"154\" style=\"vertical-align: -4px;\" \/><a id=\"figure3\"><\/a><\/p>\n<div id=\"CNX_ElemAlg_Figure_01_03_003_new\" class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\"><\/div>\n<figure style=\"width: 401px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_003_new.jpg\" alt=\"A number line extends from negative four to four. Points are plotted at negative four, negative three, negative two, negative one, zero, one, two, 3, and four.\" width=\"401\" height=\"44\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 3 All the marked numbers are called integers.<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655160945\" data-type=\"problem\">\n<p id=\"fs-id1170654952892\">Order each of the following pairs of numbers, using &lt; or &gt;: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-38826cadfc5dbf2d92abbbfd6d370410_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b5c3e12330dabaeec7413281aba0f134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e99b7c7ddd421682faf67e5b5f865882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/> d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9b93660fe4bfeaf9a69e38dd0cba4d18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655177826\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1170655192048\">It may be helpful to refer to the number line shown.<\/p>\n<p><span id=\"fs-id1170655201463\" data-type=\"media\" data-alt=\"A number line ranges from negative twenty to fifteen with ticks marks between numbers. Every fifth tick mark is labeled a number. Points are plotted at points negative twenty, negative 4, negative 1, 2, 6, 9 and 14.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_004_img_new.jpg\" alt=\"A number line ranges from negative twenty to fifteen with ticks marks between numbers. Every fifth tick mark is labeled a number. Points are plotted at points negative twenty, negative 4, negative 1, 2, 6, 9 and 14.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<table id=\"eip-672\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>a) 14 is to the right of 6 on the number line.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-38826cadfc5dbf2d92abbbfd6d370410_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b5c3e12330dabaeec7413281aba0f134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-38826cadfc5dbf2d92abbbfd6d370410_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b5c3e12330dabaeec7413281aba0f134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>b) \u22121 is to the left of 9 on the number line.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e99b7c7ddd421682faf67e5b5f865882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>\u00a0 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d6df449ad747b18616e15bc67d0f6820_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#60;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"54\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>c) \u22121 is to the right of \u22124 on the number line.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>d) 2 is to the right of \u221220 on the number line.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9b93660fe4bfeaf9a69e38dd0cba4d18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9b93660fe4bfeaf9a69e38dd0cba4d18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655162060\" data-type=\"problem\">\n<p id=\"fs-id1170654967573\">Order each of the following pairs of numbers, using &lt; or &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-da2df9cb14c5b957b4e7f8568a97d9e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#58;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"3\" style=\"vertical-align: 0px;\" \/> a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b59223d2c162a0fdd163af8597f0845e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-aca3387e59afe477960eabc2f23b3db5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-00c6d30c5f7439a21caf981437f64be1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/> d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0a516826583d9ebf41e251f45bc98b74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654978104\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654967500\">a) &gt; b) &lt; c) &gt; d) &gt;<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655114791\">You may have noticed that, on the <span class=\"no-emphasis\" data-type=\"term\">number line<\/span>, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers 2 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> are the same distance from zero, they are called opposite<strong data-effect=\"bold\">s<\/strong>. The opposite of 2 is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/>, and the opposite of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> is 2<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Opposite<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The <strong data-effect=\"bold\">opposite<\/strong> of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654959734\"><a class=\"autogenerated-content\" href=\"#figure4\">(Figure 4)<\/a> illustrates the definition.<\/p>\n<p>The opposite of 3 is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>.<a id=\"figure4\"><\/a><\/p>\n<div id=\"CNX_ElemAlg_Figure_01_03_005_new\" class=\"bc-figure figure\">\n<figure style=\"width: 440px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_005_new.jpg\" alt=\"A number line ranges from negative 4 to 4. There are two brackets above the number line. The bracket on the left spans from negative three to 0. The bracket on the right spans from zero to three. Points are plotted on both negative three and three.\" width=\"440\" height=\"81\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"fs-id1170655189167\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Opposite Notation<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170655174724\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-236972f3904fc6d64d7fafb16c1fe32d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"22\" style=\"vertical-align: 0px;\" \/> means the opposite of the number <em data-effect=\"italics\">a<\/em>.<\/p>\n<p id=\"fs-id1170655160739\">The notation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-236972f3904fc6d64d7fafb16c1fe32d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"22\" style=\"vertical-align: 0px;\" \/> is read as \u201cthe opposite of <em data-effect=\"italics\">a<\/em>.\u201d<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654920902\">The whole numbers and their opposites are called the integers. The integers are the numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-759e65d987e3039dac89a95f0411c0c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<div id=\"fs-id1170655134778\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Integers<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170654954489\">The whole numbers and their opposites are called the <strong data-effect=\"bold\">integers<\/strong>.<\/p>\n<p id=\"fs-id1170654878555\">The integers are the numbers<\/p>\n<div id=\"fs-id1170655114692\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-759e65d987e3039dac89a95f0411c0c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#49;&#44;&#50;&#44;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Simplify: Expressions with Absolute Value<\/h1>\n<p>We saw that numbers such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4690b9da1469feb61ac2499a1cdaa2bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"75\" style=\"vertical-align: -1px;\" \/> are opposites because they are the same distance from 0 on the number line. They are both two units from 0. The distance between 0 and any number on the number line is called the <strong data-effect=\"bold\">absolute value<\/strong> of that number.<\/p>\n<div id=\"fs-id1170655195725\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Absolute Value<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170655354299\">The absolute value of a number is its distance from 0 on the number line.<\/p>\n<p id=\"fs-id1170654982157\">The absolute value of a number <em data-effect=\"italics\">n<\/em> is written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b38a586dc76a31dfa818cbaca2da26f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#110;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"17\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655160915\">For example,<\/p>\n<ul id=\"fs-id1166424918316\" data-bullet-style=\"bullet\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c33bb96d6d1c81da55e768cda4296ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\" \/> units away from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-43406138e38267e27d0754c1694527ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#53;&#124;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\" \/>.<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2e1b7d211219fce780f09660c400f8b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: 0px;\" \/> units away from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6f2f975fcc9d23abe8391088e7b567a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#53;&#124;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\" \/>.<\/li>\n<\/ul>\n<p id=\"fs-id1170655134144\"><a class=\"autogenerated-content\" href=\"#figure5\">Figure 5<\/a> illustrates this idea.<\/p>\n<p>The integers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-eb86c74560a3d761e1fa66a8f0a38756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#97;&#114;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"82\" style=\"vertical-align: -1px;\" \/> units away from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5e437be25f29374d30f66cd46adf81c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>.<a id=\"figure5\"><\/a><\/p>\n<div id=\"CNX_ElemAlg_Figure_01_03_009_new\" class=\"bc-figure figure\">\n<figure style=\"width: 385px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_009_new.jpg\" alt=\"A number line is shown ranging from negative 5 to 5. A bracket labeled \u201c5 units\u201d lies above the points negative 5 to 0. An arrow labeled \u201cnegative 5 is 5 units from 0, so absolute value of negative 5 equals 5.\u201d is written above the labeled bracket. A bracket labeled \u201c5 units\u201d lies above the points \u201c0\u201d to \u201c5\u201d. An arrow labeled \u201c5 is 5 units from 0, so absolute value of 5 equals 5.\u201d and is written above the labeled bracket.\" width=\"385\" height=\"130\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1170654957170\">The absolute value of a number is never negative (because distance cannot be negative). The only number with absolute value equal to zero is the number zero itself, because the distance from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-95801e7762090eef6337adf96db73f82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/> on the number line is zero units.<\/p>\n<div id=\"fs-id1170654997376\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Property of Absolute Value<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170654916082\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b6cd98f4266f7f491a976c3270ec303d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#110;&#124;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> for all numbers<\/p>\n<p id=\"fs-id1170655074945\">Absolute values are always greater than or equal to zero!<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655150245\">Mathematicians say it more precisely, \u201cabsolute values are always non-negative.\u201d Non-negative means greater than or equal to zero.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655164890\" data-type=\"problem\">\n<p id=\"fs-id1170655164892\">Simplify:\u00a0a)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-87f810b4b629fae318134b52c7adc57f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"15\" style=\"vertical-align: -4px;\" \/>\u00a0b)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6d392a34a8207aa1aa4ac2a97dddabf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#52;&#52;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/>\u00a0c)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d3a2c72b2b6c1d226f102034f66b93ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#48;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"15\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655213327\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1170654944089\">The absolute value of a number is the distance between the number and zero. Distance is never negative, so the absolute value is never negative.<\/p>\n<p id=\"fs-id1166424887501\">a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-87f810b4b629fae318134b52c7adc57f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#51;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"15\" style=\"vertical-align: -4px;\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b32a7d27ae3987e04d57cc347c4f32cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/p>\n<p>b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6d392a34a8207aa1aa4ac2a97dddabf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#52;&#52;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-208b78e6414745ae3bc25f5465c1df5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: -1px;\" \/><\/p>\n<p>c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d3a2c72b2b6c1d226f102034f66b93ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#48;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"15\" style=\"vertical-align: -4px;\" \/><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fb9a9e4954a2381b702ec733a76df155_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655074703\" data-type=\"problem\">\n<p id=\"fs-id1170655178086\">Simplify:\u00a0a)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a97639a72b2f49a10646865ea7d7e6b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#52;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"15\" style=\"vertical-align: -4px;\" \/>\u00a0b)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c057723d9ce4e2bcbed4fdd3cb7254a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#50;&#56;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/>\u00a0c)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d3a2c72b2b6c1d226f102034f66b93ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#48;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"15\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655161378\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655161380\">a) 4 b) 28 c) 0<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654887750\">In the next example, we\u2019ll order expressions with absolute values. Remember, positive numbers are always greater than negative numbers!<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1170654953584\">Fill in &lt;, &gt;, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e675109dbfb543da39879cbe413a3064_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"37\" style=\"vertical-align: 0px;\" \/> for each of the following pairs of numbers:<\/p>\n<p id=\"fs-id1170654916476\">a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-29f80c98885a0c2b74a1fc1f5e6d86b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#53;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-545b962a2ddeb7bc62f8baebcaa3d5b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#124;&#45;&#53;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-23212a120eb92d226d44696a0b80bead_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b5c22585d424149a2f06385206336c56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#124;&#45;&#56;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\" \/>c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f7db1529cce29849fd2d8a54b0fa1440_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#124;&#45;&#57;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\" \/> d) &#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-60f20f07f8f7c62331f2d8fe388a61b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-97965dd09e44097632b9a81977b96c4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#124;&#45;&#49;&#54;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1170655106909\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\">\n<table style=\"border-collapse: collapse; width: 100%; height: 184px;\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-29f80c98885a0c2b74a1fc1f5e6d86b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#53;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/>___ &#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-29f80c98885a0c2b74a1fc1f5e6d86b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#53;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 30px;\">\n<td>a) Simplify.<br \/>\nOrder.<\/td>\n<td>5 ___ -5<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td><\/td>\n<td>5\u00a0 &gt; -5<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-29f80c98885a0c2b74a1fc1f5e6d86b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#53;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/>\u00a0 &gt; &#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-29f80c98885a0c2b74a1fc1f5e6d86b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#53;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td>b) Simplify.<br \/>\nOrder.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-23212a120eb92d226d44696a0b80bead_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> ___ &#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4edbd13afbf91b71c1d89205f3d7869d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#56;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td><\/td>\n<td>8 ___ -8<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td><\/td>\n<td>8 &gt; -8<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td><\/td>\n<td>8 &gt; &#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4edbd13afbf91b71c1d89205f3d7869d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#56;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td>c) Simplify.<br \/>\nOrder.<\/td>\n<td>9 ___ &#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c4d077845d89c58590a6bd9ef094faa4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#57;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\" \/><\/p>\n<p>-9 ___ -9<\/p>\n<p>-9 = -9<\/p>\n<p>-9 = &#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c4d077845d89c58590a6bd9ef094faa4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#57;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td>d) Simplify.<br \/>\nOrder.<\/td>\n<td>&#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-890d448b349cb30ba797bda0ae641dbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#45;&#49;&#54;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"44\" style=\"vertical-align: -5px;\" \/> ___ &#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-40fd066ba59d22d02733456214eb81cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#49;&#54;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"46\" style=\"vertical-align: -5px;\" \/><\/p>\n<p>16 ____ -16<\/p>\n<p>16 &gt; -16<\/p>\n<p>&#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-890d448b349cb30ba797bda0ae641dbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#45;&#49;&#54;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"44\" style=\"vertical-align: -5px;\" \/> &gt; &#8211;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-40fd066ba59d22d02733456214eb81cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#49;&#54;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"46\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655121090\" data-type=\"problem\">\n<p id=\"fs-id1170655099637\">Fill in &lt;, &gt;, or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b8690b7efd237bfe32a6e92e3b699b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"4\" width=\"13\" style=\"vertical-align: 2px;\" \/> for each of the following pairs of numbers:\u00a0a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c4d077845d89c58590a6bd9ef094faa4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#57;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\" \/>___-<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c4d077845d89c58590a6bd9ef094faa4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#57;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/>___-<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7f109d5404c977d1a4cd37023190a812_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#50;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4edbd13afbf91b71c1d89205f3d7869d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#56;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\" \/><br \/>\nd) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cb4cc8cbcb582b264a149b5a50dd8df6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#45;&#40;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"29\" style=\"vertical-align: -5px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f7db1529cce29849fd2d8a54b0fa1440_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#124;&#45;&#57;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655228722\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655352028\">a) &gt; b) &gt; c) &lt; d) &gt;<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655060167\">We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the <span class=\"no-emphasis\" data-type=\"term\">absolute value<\/span> of the resulting number.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<caption>Grouping Symbols<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 50%;\">Parentheses<\/td>\n<td style=\"width: 50%;\">( )<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Brackets<\/td>\n<td style=\"width: 50%;\">[ ]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Braces<\/td>\n<td style=\"width: 50%;\">{ }<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Absolute value<\/td>\n<td style=\"width: 50%;\">| |<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170655103794\">In the next example, we simplify the expressions inside absolute value bars first, just like we do with parentheses.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655218379\" data-type=\"problem\">\n<p id=\"fs-id1170655000668\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7d8edc5774e4274d735ff981d931a1f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#45;&#124;&#49;&#57;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"155\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654968392\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-46\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7d8edc5774e4274d735ff981d931a1f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#45;&#124;&#49;&#57;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"155\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Work inside parentheses first: subtract 2 from 6.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5b2eba697acad7bccc802b9eb876a5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#45;&#124;&#49;&#57;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply 3(4).<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6009efcc047f5b92d5a99dc4f37f5346_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#45;&#124;&#49;&#57;&#45;&#49;&#50;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Subtract inside the absolute value bars.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-43d89b4f0f228bd9a08f73f516854081_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#45;&#124;&#55;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Take the absolute value.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a5eaef3ba285fc7d0ae3dfdd9efe6149_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"48\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b976c8c10558ac50c42d953f3ee38707_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655222089\" data-type=\"problem\">\n<p id=\"fs-id1170655222091\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-092df2527fd0d339aadc373741e4f2dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#57;&#45;&#124;&#49;&#49;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"154\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654963377\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654963379\">16<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Add Integers<\/h1>\n<p id=\"fs-id1170654942147\">Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.<\/p>\n<p id=\"fs-id1170654982815\">We will use two colour counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.<\/p>\n<p id=\"fs-id1170654940836\">We let one colour (blue) represent positive. The other colour (red) will represent the negatives. If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.<\/p>\n<p><span id=\"fs-id1170654940841\" data-type=\"media\" data-alt=\"In this image we have a blue counter above a red counter with a circle around both. The equation to the right is 1 plus negative 1 equals 0.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_014_img_new.jpg\" alt=\"In this image we have a blue counter above a red counter with a circle around both. The equation to the right is 1 plus negative 1 equals 0.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1170654925278\">We will use the counters to show how to add the four addition facts using the numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5a46791a391dd60ec10b6fdd529ca8ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#44;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"39\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c8f99a729c7a37f3083f2bbfa4f2a1af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#44;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"40\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3a5cb80ca32bb44bfa32aca2af8b6463_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"386\" style=\"vertical-align: -4px;\" \/><\/p>\n<p>The first example adds 5 positives and 3 positives\u2014both positives.<\/p>\n<p>The second example adds 5 negatives and 3 negatives\u2014both negatives.<\/p>\n<p id=\"fs-id1170654905046\">In each case we got 8\u2014either 8 positives or 8 negatives.<\/p>\n<p id=\"fs-id1170654905109\">When the signs were the same, the counters were all the same color, and so we added them.<\/p>\n<p><span id=\"fs-id1170655040356\" data-type=\"media\" data-alt=\"This figure is divided into two columns. In the left column there are eight blue counters in a horizontal row. Under them is the text \u201c8 positives.\u201d Centred under this is the equation 5 plus 3 equals 8. In the right column are eight red counters in a horizontal row which are labled below with the phrase \u201c8 negatives\u201d. Centred under this is the equation negative 5 plus negative 3 equals negative 8, where negative 3 is in parentheses.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_021_img_new.jpg\" alt=\"This figure is divided into two columns. In the left column there are eight blue counters in a horizontal row. Under them is the text \u201c8 positives.\u201d Centred under this is the equation 5 plus 3 equals 8. In the right column are eight red counters in a horizontal row which are labled below with the phrase \u201c8 negatives\u201d. Centred under this is the equation negative 5 plus negative 3 equals negative 8, where negative 3 is in parentheses.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1170655030242\">So what happens when the signs are different? Let\u2019s add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9db94c4517228eb349cd4426ad66e71b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"52\" style=\"vertical-align: -2px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cd386991eb516c110c242436558446ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p id=\"fs-id1170655057773\">When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative.<\/p>\n<p><span id=\"fs-id1170655058518\" data-type=\"media\" data-alt=\"Two images are shown and labeled. The left image shows five red counters in a horizontal row drawn above three blue counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written \u201cnegative 5 plus 3\u201d and below is written \u201cMore negatives \u2013 the sum is negative.\u201d The right image shows five blue counters in a horizontal row drawn above three red counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written \u201c5 plus negative 3\u201d and below is written \u201cMore positives \u2013 the sum is positive.\u201d\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_026_img_new.jpg\" alt=\"Two images are shown and labeled. The left image shows five red counters in a horizontal row drawn above three blue counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written \u201cnegative 5 plus 3\u201d and below is written \u201cMore negatives \u2013 the sum is negative.\u201d The right image shows five blue counters in a horizontal row drawn above three red counters in a horizontal row, where the first three pairs of red and blue counters are circled. Above this diagram is written \u201c5 plus negative 3\u201d and below is written \u201cMore positives \u2013 the sum is positive.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655058550\" data-type=\"problem\">\n<p id=\"fs-id1170655058552\">Add: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1211b5f02078795082d5e423b624cad8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"51\" style=\"vertical-align: -2px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b070b9d885ab879390b3c67d95240032_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655029787\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1166422829183\">a)<\/p>\n<table id=\"eip-id1169754360753\" style=\"width: 100%;\" summary=\"In this image we have two columns. The top of the right column has the equation \u201cnegative 1 plus 5.\u201d The next line down in the left column reads, \u201cThere are more positives, so the sum is positive.\u201d Next to this in the right column are five blue counters. Under this, one red counter. The first blue counter and the single red counter are circled. Under this in the same column it says \u201c4.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"center\">\u22121 + 5<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1169749528166\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_027_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>There are more positives, so the sum is positive.<\/td>\n<td data-align=\"center\">4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>b)<\/p>\n<table id=\"eip-id1169750608265\" style=\"width: 100%;\" summary=\"In this image we have two columns. The top of the right column has the equation \u201c1 plus negative 5.\u201d The next line down in the left column reads, \u201cThere are more negatives, so the sum is negative.\u201d Next to this in the right column is one blue counter. Under this, five red counters. The single blue counter and the first of the five red counters is circled. Under this in the same column it says \u201cnegative 4.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"center\">1 + (\u22125)<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span id=\"eip-id1169750608293\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_028_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>There are more negatives, so the sum is negative.<\/td>\n<td data-align=\"center\">\u22124<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654904893\" data-type=\"problem\">\n<p id=\"fs-id1170654904895\">Add:\u00a0a)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6a4dcc55080d931ad32225aa3a323aca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"52\" style=\"vertical-align: -2px;\" \/>\u00a0b)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2251ca4270ec721714b863c4dbed86f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655066666\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655066668\">a) 2 b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655066171\">Now that we have added small positive and negative integers with a model, we can visualize the model in our minds to simplify problems with any numbers.<\/p>\n<p id=\"fs-id1170655067182\">When you need to add numbers such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bfa527ceb2ba44255562b4665edb273b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#55;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" style=\"vertical-align: -5px;\" \/>, you really don\u2019t want to have to count out 37 blue counters and 53 red counters. With the model in your mind, can you visualize what you would do to solve the problem?<\/p>\n<p id=\"fs-id1170655067203\">Picture 37 blue counters with 53 red counters lined up underneath. Since there would be more red (negative) counters than blue (positive) counters, the sum would be <em data-effect=\"italics\">negative<\/em>. How many more red counters would there be? Because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b09d319d46b6d8d70bfc4295bb36ed39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#51;&#45;&#51;&#55;&#61;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"99\" style=\"vertical-align: 0px;\" \/>, there are 16 more red counters.<\/p>\n<p id=\"fs-id1170655067279\">Therefore, the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bfa527ceb2ba44255562b4665edb273b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#55;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" style=\"vertical-align: -5px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-60f20f07f8f7c62331f2d8fe388a61b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<div id=\"fs-id1170655067436\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b09dba2d0eb6159497a0515ef08f5bfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#55;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"140\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1170655069654\">Let\u2019s try another one. We\u2019ll add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-98f58f6d65f00c5cded6b2d78160a1ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\" \/>. Again, imagine 74 red counters and 27 more red counters, so we\u2019d have 101 red counters. This means the sum is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bef0630181569824ae99bb7fcbc6cc43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-32463a2e4155942839d70fc5b66fd3b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#45;&#49;&#48;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"161\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1170655069404\">Let\u2019s look again at the results of adding the different combinations of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5a46791a391dd60ec10b6fdd529ca8ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#44;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"39\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c8f99a729c7a37f3083f2bbfa4f2a1af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#44;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"40\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Addition of Positive and Negative Integers<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655069465\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bef4f6998ff330d86687184477372218_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#111;&#116;&#104;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#44;&#32;&#115;&#117;&#109;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#111;&#116;&#104;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#44;&#32;&#115;&#117;&#109;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"477\" style=\"vertical-align: -25px;\" \/><\/div>\n<p id=\"fs-id1170655072721\">When the signs are the same, the counters would be all the same color, so add them.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0ad585088aeed787d8a28255a8db63cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#116;&#32;&#115;&#105;&#103;&#110;&#115;&#44;&#32;&#109;&#111;&#114;&#101;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#115;&#44;&#32;&#115;&#117;&#109;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#116;&#32;&#115;&#105;&#103;&#110;&#115;&#44;&#32;&#109;&#111;&#114;&#101;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#115;&#44;&#32;&#115;&#117;&#109;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"732\" style=\"vertical-align: -25px;\" \/><\/p>\n<p id=\"fs-id1170655163407\">When the signs are different, some of the counters would make neutral pairs, so subtract to see how many are left.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654942464\">Visualize the model as you simplify the expressions in the following examples.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654932186\" data-type=\"problem\">\n<p id=\"fs-id1170654932188\">Simplify:\u00a0a)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-214c9b1a981dd7bff53ffa89ff9c703c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#57;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/>\u00a0b)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-95d33e01b1413af803051b0d9cc018be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654942539\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<ol id=\"fs-id1166422639606\" class=\"circled\" type=\"a\">\n<li>\u00a0Since the signs are different, we subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8ed34c712e502aaff1fa3497a37fcfb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#57;&#32;&#102;&#114;&#111;&#109;&#32;&#52;&#55;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"86\" style=\"vertical-align: -1px;\" \/> The answer will be negative because there are more negatives than positives.<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7800ea26f03c2ce5a096bbfa4c067c98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#57;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#100;&#100;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#50;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"176\" style=\"vertical-align: -12px;\" \/><\/li>\n<li>\u00a0Since the signs are the same, we add. The answer will be negative because there are only negatives.<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d19bd2ae53ecc454ac47c49561c5f9a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#100;&#100;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"185\" style=\"vertical-align: -12px;\" \/><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654941250\" data-type=\"problem\">\n<p id=\"fs-id1170654941252\">Simplify:\u00a0a)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1fe9638fae9c4a1075a1341898477240_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#49;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -4px;\" \/>\u00a0b)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3bcd2a87adedafbd2452193e94c40563_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654941827\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655215496\">a)<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-862746467ef7a474e7ee68e5f343ac7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/> b)<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0a516826583d9ebf41e251f45bc98b74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654944039\">The techniques used up to now extend to more complicated problems, like the ones we\u2019ve seen before. Remember to follow the order of operations!<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654929156\" data-type=\"problem\">\n<p id=\"fs-id1170654929159\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-448672ff0a1ac55cbe96ad8d96b0b670_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654934976\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-256\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-448672ff0a1ac55cbe96ad8d96b0b670_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify inside the parentheses.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8be44f2d6976b8defb59f8d2a28324be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#43;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-59c7a6beeb76118dfdb9dcdde558098a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"60\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add left to right.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5450913cc453faf132dc56e5965ca797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654936188\" data-type=\"problem\">\n<p id=\"fs-id1170654936190\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a13f9dcea68de157b63e9ecf5b06c754_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#43;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"121\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654944107\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654944110\">13<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<h1>Subtract Integers<\/h1>\n<p id=\"fs-id1170654921208\">We will continue to use counters to model the subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.<\/p>\n<p id=\"fs-id1170654921213\">Perhaps when you were younger, you read <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1c2458401e1ba61cb79b5e0d0af7783f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#96;&#96;&#125;&#53;&#45;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#39;&#39;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: 0px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-562942cc14b3e1da3b517d2a107b8a5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#96;&#96;&#125;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"15\" style=\"vertical-align: 0px;\" \/> take away <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1a0d80aaea08b754c67895d1d8974745_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#92;&#116;&#101;&#120;&#116;&#123;&#39;&#39;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> When you use counters, you can think of subtraction the same way!<\/p>\n<p id=\"fs-id1170654932154\">We will model the four subtraction facts using the numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4a1d3ea4963f568cabd97329456036b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<div id=\"fs-id1170654936672\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-931460af5fd03a39bd83c51d7b4bb9df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"386\" style=\"vertical-align: -4px;\" \/><\/div>\n<p>The first example, we subtract 3 positives from 5 positives and end up with 2 positives.<\/p>\n<p id=\"fs-id1170654942784\">In the second example, we subtract 3 negatives from 5 negatives and end up with 2 negatives.<\/p>\n<p id=\"fs-id1170654942787\">Each example used counters of only one color, and the \u201ctake away\u201d model of subtraction was easy to apply.<\/p>\n<p><span id=\"fs-id1170654942792\" data-type=\"media\" data-alt=\"Two images are shown and labeled. The first image shows five blue counters, three of which are circled with an arrow. Above the counters is the equation \u201c5 minus 3 equals 2.\u201d The second image shows five red counters, three of which are circled with an arrow. Above the counters is the equation \u201cnegative 5, minus, negative 3, equals negative 2.\u201d\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_031_img_new.jpg\" alt=\"Two images are shown and labeled. The first image shows five blue counters, three of which are circled with an arrow. Above the counters is the equation \u201c5 minus 3 equals 2.\u201d The second image shows five red counters, three of which are circled with an arrow. Above the counters is the equation \u201cnegative 5, minus, negative 3, equals negative 2.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1170654935830\">What happens when we have to subtract one positive and one negative number? We\u2019ll need to use both white and red counters as well as some <span class=\"no-emphasis\" data-type=\"term\">neutral pairs<\/span>. Adding a neutral pair does not change the value. It is like changing quarters to nickels\u2014the value is the same, but it looks different.<\/p>\n<ul id=\"fs-id1166424901740\" data-bullet-style=\"bullet\">\n<li>To subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a029da03c39ec48a0d5bebd17696a97b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: 0px;\" \/>, we restate it as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/> take away 3.<\/li>\n<\/ul>\n<p id=\"fs-id1170654928179\">We start with 5 negatives. We need to take away 3 positives, but we do not have any positives to take away.<\/p>\n<p id=\"fs-id1170654928184\">Remember, a neutral pair has value zero. If we add 0 to 5 its value is still 5. We add neutral pairs to the 5 negatives until we get 3 positives to take away.<\/p>\n<table id=\"eip-id1169752192784\" style=\"width: 100%;\" summary=\"In this image, we have two columns and six rows. The first row in the right column reads, \u201cnegative 5 minus 3 means negative 5 take away 3.\u201d The next row down on the left reads, \u201cWe started with five negatives.\u201d Next to this are five red counters in a horizontal row with \u201cnegative\u201d five underneath. The next row reads, \u201cWe now add the neutrals needed to get three positives. Next to this in the right column are five red counters, a space, and three red counters. Under the three red counters are three blue counters. The next row down reads, \u201cWe remove the three positives. Next to this in the right column are the five red counters, a space, and three red counters. Under the three red counters are three blue counters with a circled arrow around them. The next row down reads, \u201cwe are left with eight negatives.\u201d Next to this in the right column in a horizontal row are five red counters, a space, followed by three red counters. Under the counters reads \u201c8 negatives\u201d. The last row in the left column reads, \u201cThe difference of negative 5 and 3 is negative 8.\u201d Next to this in the right column is the equation, \u201cnegative 5 minus 3equals negative eight.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\u22125 \u2212 3 means \u22125 take away 3.<\/td>\n<\/tr>\n<tr>\n<td>We start with 5 negatives.<\/td>\n<td><span id=\"eip-id1169754016089\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_032a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>We now add the neutrals needed to get 3 positives.<\/td>\n<td><span id=\"eip-id1169754016106\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_032b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>We remove the 3 positives.<\/td>\n<td><span id=\"eip-id1169754016122\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_032c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>We are left with 8 negatives.<\/td>\n<td><span id=\"eip-id1169754361424\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_032d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>The difference of \u22125 and 3 is \u22128.<\/td>\n<td>\u22125 \u2212 3 = \u22128<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170655261587\">And now, the fourth case, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f95d94f6a2040d6be872ba37f3614152_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/>. We start with 5 positives. We need to take away 3 negatives, but there are no negatives to take away. So we add neutral pairs until we have 3 negatives to take away.<\/p>\n<table id=\"eip-id1169754361484\" style=\"width: 100%;\" summary=\"In this image, we have two columns and six rows. The first row in the right column reads, \u201c5 minus negative 3 means 5 take away negative 3.\u201d The next row down on the left reads, \u201cWe started with five positives.\u201d Next to this are five blue counters in a horizontal row. The next row reads, \u201cWe now add the needed neutral pairs. Next to this in the right column are five blue counters, a space, and three blue counters. Under the three blue counters are three red counters. The next row down reads, \u201cWe remove the 3 negatives. Next to this in the right column are the five blue counters, a space, and three blue counters. Under the three blue counters are three red counters with a circled arrow around them. The next row down reads, \u201cWe are left with eight positives.\u201d Next to this in the right column in a horizontal row are five blue counters, a space, followed by three blue counters. Under the counters reads \u201c8 positives\u201d. The last row in the left column reads, \u201cThe difference of 5 and negative 3 is 8.\u201d Next to this in the right column is the equation, \u201c5 minus negative 3equals eight.\u201d\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>5 \u2212 (\u22123) means 5 take away \u22123.<\/td>\n<\/tr>\n<tr>\n<td>We start with 5 positives.<\/td>\n<td><span id=\"eip-id1169753968359\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_033a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>We now add the needed neutrals pairs.<\/td>\n<td><span id=\"eip-id1169753968375\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_033b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>We remove the 3 negatives.<\/td>\n<td><span id=\"eip-id1169753968392\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_033c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>We are left with 8 positives.<\/td>\n<td><span id=\"eip-id1169752181606\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_033d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>The difference of 5 and \u22123 is 8.<\/td>\n<td>5 \u2212 (\u22123) = 8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"example\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654941998\" data-type=\"problem\">\n<p id=\"fs-id1170654942000\">Subtract:\u00a0a)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b1d1c3eed482bbd31339b691e61cf6ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/>\u00a0b)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b5b23456808ccbcc7fc4f81a2a5bac91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654943757\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<p id=\"fs-id1167829621734\">a)<\/p>\n<table id=\"eip-id1169749482384\" style=\"width: 100%;\" summary=\"Part \u201ca\u201d has three red counters, a space, and one red counter in a horizontal row. Next to this on the right is the equation \u201cnegative 3 minus 1.\u201d The next line down on the left reads,\u201dTake 1 positive from the one added neutral pair\u201d. Next to this on the right is a blue counter circled with an arrow. Next to this reads \u201cnegative 4.\u201d Part \u201cb\u201d has three blue counters, a space, and one blue counter in a horizontal row. Next to this on the right is the equation \u201c3 minus negative 1.\u201d The next line down on the left reads, \u201cTake one negative from the one added neutral pair.\u201d Next to this on the right is a red counter circled with an arrow. Next to this reads \u201c4\u201d.\" data-label=\"\">\n<tbody>\n<tr>\n<td>Take 1 positive from the one added neutral pair.<\/td>\n<td><span id=\"eip-id1169749482408\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_034a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><br \/>\n<span id=\"eip-id1169749482428\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_034b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<td data-valign=\"top\">\u22123 \u2212 1<\/p>\n<p>\u22124<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836289563\">b)<\/p>\n<table id=\"fs-id1167824767322\" style=\"width: 100%;\" summary=\"Part \u201cb\u201d has three blue counters, a space, and one blue counter in a horizontal row. Next to this on the right is the equation \u201c3 minus negative 1.\u201d The next line down on the left reads, \u201cTake one negative from the one added neutral pair.\u201d Next to this on the right is a red counter circled with an arrow. Next to this reads \u201c4\u201d.\" data-label=\"\">\n<tbody>\n<tr>\n<td>Take 1 negative from the one added neutral pair.<\/td>\n<td><span id=\"eip-id1169749482451\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_034c_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><br \/>\n<span id=\"eip-id1169749482475\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_034d_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<td data-align=\"center\" data-valign=\"top\">3 \u2212 (\u22121)<\/p>\n<p>4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654924800\" data-type=\"problem\">\n<p id=\"fs-id1170654941927\">Subtract:\u00a0a)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b6302a9df0c3462c19601388a3e35158_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/>\u00a0b)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-50136d0783a08d899f96aa2473b2eb31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654942432\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654942434\">a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-24329a36a4e10288288979f77a565ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> b) 10<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p>Have you noticed that <em data-effect=\"italics\">subtraction of signed numbers can be done by adding the opposite<\/em>? In <a class=\"autogenerated-content\" href=\"#fs-id1170654941998\">Example 8<\/a>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b1d1c3eed482bbd31339b691e61cf6ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/> is the same as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-75db80a14417df8d60e8ec6d03bb6f06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b5b23456808ccbcc7fc4f81a2a5bac91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/> is the same as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0086024923882694cf8386d01386dde5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"39\" style=\"vertical-align: -2px;\" \/>. You will often see this idea, the subtraction property, written as follows:<\/p>\n<\/div>\n<div id=\"fs-id1170654903656\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Subtraction Property<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654903661\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2500da82de4ecc9460ada1e9df0cd6dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#45;&#98;&#61;&#97;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"equation\" data-label=\"\"><\/div>\n<div class=\"unnumbered\" data-type=\"equation\" data-label=\"\">Subtracting a number is the same as adding its opposite.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654903717\">Look at these two examples.<\/p>\n<p><span id=\"fs-id1170654962580\" data-type=\"media\" data-alt=\"Two images are shown and labeled. The first image shows four gray spheres drawn next to two gray spheres, where the four are circled in red, with a red arrow leading away to the lower left. This drawing is labeled above as \u201c6 minus 4\u201d and below as \u201c2.\u201d The second image shows four gray spheres and four red spheres, drawn one above the other and circled in red, with a red arrow leading away to the lower left, and two gray spheres drawn to the side of the four gray spheres. This drawing is labeled above as \u201c6 plus, open parenthesis, negative 4, close parenthesis\u201d and below as \u201c2.\u201d\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_035_img_new.jpg\" alt=\"Two images are shown and labeled. The first image shows four gray spheres drawn next to two gray spheres, where the four are circled in red, with a red arrow leading away to the lower left. This drawing is labeled above as \u201c6 minus 4\u201d and below as \u201c2.\u201d The second image shows four gray spheres and four red spheres, drawn one above the other and circled in red, with a red arrow leading away to the lower left, and two gray spheres drawn to the side of the four gray spheres. This drawing is labeled above as \u201c6 plus, open parenthesis, negative 4, close parenthesis\u201d and below as \u201c2.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<div id=\"fs-id1166424959329\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-912270a66466a24c6ced13bfec698783_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#45;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#105;&#118;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#32;&#97;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#54;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"306\" style=\"vertical-align: -5px;\" \/>.<\/div>\n<p id=\"fs-id1170654962732\">Of course, when you have a subtraction problem that has only positive numbers, like <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d5998bc2d047a935f18b7f8b1bead811_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/>, you just do the subtraction. You already knew how to subtract <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d5998bc2d047a935f18b7f8b1bead811_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/> long ago. But <em data-effect=\"italics\">knowing<\/em> that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d5998bc2d047a935f18b7f8b1bead811_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/> gives the same answer as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-599bd531dc22ff1a8c27da9267a11c3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/> helps when you are subtracting negative numbers. Make sure that you understand how <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d5998bc2d047a935f18b7f8b1bead811_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-599bd531dc22ff1a8c27da9267a11c3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/> give the same results!<\/p>\n<p id=\"fs-id1170655075753\">Look at what happens when we subtract a negative.<\/p>\n<p><span id=\"fs-id1170655075756\" data-type=\"media\" data-alt=\"This figure is divided vertically into two halves. The left part of the figure contains the expression 8 minus negative 5, where negative 5 is in parentheses. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the group of five blue counters is a group of five red counters, which are circled. The circle has an arrow pointing away toward bottom left of the image, symbolizing subtraction. Below the counters is the number 13. The right part of the figure contains the expression 8 plus 5. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the counters is the number 13.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_03_036_img_new.jpg\" alt=\"This figure is divided vertically into two halves. The left part of the figure contains the expression 8 minus negative 5, where negative 5 is in parentheses. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the group of five blue counters is a group of five red counters, which are circled. The circle has an arrow pointing away toward bottom left of the image, symbolizing subtraction. Below the counters is the number 13. The right part of the figure contains the expression 8 plus 5. The expression sits above a group of 8 blue counters next to a group of five blue counters in a row, with a space between the two groups. Underneath the counters is the number 13.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<div id=\"fs-id1166424830896\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-df6027bd25c49d4e73342911fadc3a23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#105;&#118;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#97;&#110;&#115;&#119;&#101;&#114;&#32;&#97;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"309\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1170655077076\"><em data-effect=\"italics\">Subtracting a negative number is like adding a positive!<\/em><\/p>\n<p id=\"fs-id1170655076379\">You will often see this written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bc2443b6ace73e73410ffb046fb33f86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#97;&#43;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<p id=\"fs-id1170654962473\">What happens when there are more than three integers? We just use the order of operations as usual.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655084026\" data-type=\"problem\">\n<p id=\"fs-id1170655084028\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5bfb5f0bf8c4079f491cc6417a833b49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655083647\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-942\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5bfb5f0bf8c4079f491cc6417a833b49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Simplify inside the parentheses first.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bf972ec3e8e35cb44cc7225bbc3e7154_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Subtract left to right.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6404bcd9f623bc44bff5de5a6d38ff60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8995e2084120c1c1f4b53f490d281bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170655091566\" data-type=\"problem\">\n<p id=\"fs-id1170655091568\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-72b526e3b44869aac5ad9869f11658bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170655090951\" data-type=\"solution\">\n<details open=\"open\">\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170655090953\">3<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170655091187\" class=\"media-2\" data-type=\"note\">\n<h1>Multiply Integers<\/h1>\n<p id=\"fs-id1170654047593\">Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show <span class=\"no-emphasis\" data-type=\"term\">multiplication<\/span> of <span class=\"no-emphasis\" data-type=\"term\">integers<\/span>. Let\u2019s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.<\/p>\n<p id=\"fs-id1170654027710\">We remember that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3c7ce7788e60e03d2a0376725734513b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/> means add <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em> times. Here, we are using the model just to help us discover the pattern.<\/p>\n<p><span id=\"fs-id1170652629325\" data-type=\"media\" data-alt=\"Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads \u201cadd 5, 3 times.\u201d Below this depicts three rows of blue counters, with five counters in each row. Under this, it says \u201c15 positives.\u201d Under thisis the equation\u201c5 times 3 equals 15.\u201d The image on the right reads \u201cnegative 5 times three. The three is in parentheses. Below this it reads, \u201cadd negative five, three times.\u201d Under this are fifteen red counters in three rows of five. Below this it reads\u201d \u201c15 negatives\u201d. Below this is the equation negative five times 3 equals negative 15.\u201d\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_04_001_img_new.jpg\" alt=\"Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads \u201cadd 5, 3 times.\u201d Below this depicts three rows of blue counters, with five counters in each row. Under this, it says \u201c15 positives.\u201d Under thisis the equation\u201c5 times 3 equals 15.\u201d The image on the right reads \u201cnegative 5 times three. The three is in parentheses. Below this it reads, \u201cadd negative five, three times.\u201d Under this are fifteen red counters in three rows of five. Below this it reads\u201d \u201c15 negatives\u201d. Below this is the equation negative five times 3 equals negative 15.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1170654235984\">The next two examples are more interesting.<\/p>\n<p id=\"fs-id1171791114068\">What does it mean to multiply 5 by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-80cc5c6a8396c9c51a989af972cef479_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/> It means subtract 5, 3 times. Looking at subtraction as \u201ctaking away,\u201d it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the work space. Then we take away 5 three times.<\/p>\n<p><span id=\"fs-id1170654116877\" data-type=\"media\" data-alt=\"This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/06\/CNX_ElemAlg_Figure_01_04_002_img_new.jpg\" alt=\"This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1170654009263\">In summary:<\/p>\n<div id=\"fs-id1170653916873\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e5c974bb2de5b919a80e070a1666c45e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"328\" style=\"vertical-align: -16px;\" \/><\/div>\n<p id=\"fs-id1170653885621\">Notice that for multiplication of two signed numbers, when the:<\/p>\n<ul id=\"fs-id1166425881422\" data-bullet-style=\"bullet\">\n<li>signs are the <em data-effect=\"italics\">same<\/em>, the product is <em data-effect=\"italics\">positive<\/em>.<\/li>\n<li>signs are <em data-effect=\"italics\">different<\/em>, the product is <em data-effect=\"italics\">negative<\/em>.<\/li>\n<\/ul>\n<p id=\"fs-id1170653867968\">We\u2019ll put this all together in the chart below<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Multiplication of Signed Numbers<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For multiplication of two signed numbers:<\/p>\n<table class=\"grid\" style=\"width: 100%;\">\n<tbody>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\">Same signs<\/th>\n<td><strong>Product<\/strong><\/td>\n<td><strong>Example<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Two positives<br \/>\nTwo negatives<\/td>\n<td>Positive<br \/>\nPositive<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e6998895f6cc95e93320c3ac8d4e80ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#92;&#99;&#100;&#111;&#116;&#52;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"125\" style=\"vertical-align: -16px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"grid\" style=\"width: 100%;\">\n<tbody>\n<tr>\n<td><strong>Different signs<\/strong><\/td>\n<td><strong>Product<\/strong><\/td>\n<td><strong>Example<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Positive \\cdot negative<br \/>\nNegative \\cdot positive<\/td>\n<td>Negative<br \/>\nNegative<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8a8bde3e9cb5aa1a04adf7808eb14fb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#54;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#92;&#99;&#100;&#111;&#116;&#49;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#53;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"130\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654016481\" data-type=\"problem\">\n<p id=\"fs-id1170653756055\">Multiply: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-de7c632208e88535c1777d8c3eea9b90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;&#92;&#99;&#100;&#111;&#116;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-777d7e03559de57012fa6bee17017df6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -4px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-70eabcb64a7d9494be9d191cbf23d11d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-df846a36f642b6bb2be4b32f33f16369_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#99;&#100;&#111;&#116;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654013170\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-959\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>a)<br \/>\nMultiply, noting that the signs are different so the product is negative.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e77142885052948c1ad2369080360d8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#57;&#92;&#99;&#100;&#111;&#116;&#51;&#92;&#92;&#32;&#45;&#50;&#55;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"44\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>b)<br \/>\nMultiply, noting that the signs are the same so the product is positive.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-288b5020f6ce0368a272332f7f054c5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#49;&#48;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"61\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>c)<br \/>\nMultiply, with different signs.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5e93a5f4831eda5c57265e0b72c0543f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#45;&#51;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"48\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>d)<br \/>\nMultiply, with same signs.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-36eedafad1fb35dfb4e5deb1fdcde93e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#55;&#92;&#99;&#100;&#111;&#116;&#54;&#92;&#92;&#32;&#52;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"31\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654150864\" data-type=\"problem\">\n<p id=\"fs-id1170653870759\">Multiply: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-753d2b8d68d83e3bbc10f04cace32aa2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#92;&#99;&#100;&#111;&#116;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bc440a9bd341434fbc31c0a84282560e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -4px;\" \/> c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b361129d452a3d90c0bec2bb004052be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/> d) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f7778de7e8225296a7b3ab2b07c899a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#99;&#100;&#111;&#116;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654034104\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170653885313\">a)<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-08ed8320699be2770837da731d65c7c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> b) 28 c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c081a4d1588de9f9de1dd7d5282b4c9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/> d) 60<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170654150913\">When we multiply a number by 1, the result is the same number. What happens when we multiply a number by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>? Let\u2019s multiply a positive number and then a negative number by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/> to see what we get.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-51234b5987c36f317662fc30ae7fcc20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#99;&#100;&#111;&#116;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#116;&#104;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#116;&#104;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#51;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"528\" style=\"vertical-align: -26px;\" \/><\/p>\n<p id=\"fs-id1170652622975\">Each time we multiply a number by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/>, we get its opposite!<\/p>\n<div id=\"fs-id1170653776943\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Multiplication by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-160fe5d9875697871f0f4bbf770cd795_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#97;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"70\" style=\"vertical-align: 0px;\" \/><\/p>\n<p>Multiplying a number by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/> gives its opposite.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654068942\" data-type=\"problem\">\n<p id=\"fs-id1170654036219\">Multiply: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4cc02b0dea58170ed1d677175e755a62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#92;&#99;&#100;&#111;&#116;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: 0px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3d008c149a4735bade76f09f84da99b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653910210\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-228\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>a)<br \/>\nMultiply, noting that the signs are different so the product is negative.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cb90ab6d1f527080952f45666bef6432_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#49;&#92;&#99;&#100;&#111;&#116;&#55;&#92;&#92;&#32;&#45;&#55;&#92;&#92;&#32;&#45;&#55;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#116;&#104;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#55;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"175\" style=\"vertical-align: -26px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>b)<br \/>\nMultiply, noting that the signs are the same so the product is positive.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5a39f60ffd303d28baf399e762632ec6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#49;&#49;&#92;&#92;&#32;&#49;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#116;&#104;&#101;&#32;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#32;&#111;&#102;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#49;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"201\" style=\"vertical-align: -25px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654073784\" data-type=\"problem\">\n<p id=\"fs-id1170653851085\">Multiply: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7ca2da17fc7236aab9ecd4bc8933058a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#92;&#99;&#100;&#111;&#116;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4e5deed80daf9b22b8a48b2e79dafc77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"79\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654014381\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654231240\">a)<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> b) 17<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Divide Integers<\/h1>\n<p id=\"fs-id1170654029961\">What about <span class=\"no-emphasis\" data-type=\"term\">division<\/span>? Division is the inverse operation of multiplication. So, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-259fda24381aa07d31aa2faddde3156e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#92;&#100;&#105;&#118;&#32;&#51;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"79\" style=\"vertical-align: -1px;\" \/> because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5b53d8cea1b2e23d1bbfe4264a550e01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#99;&#100;&#111;&#116;&#51;&#61;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"71\" style=\"vertical-align: 0px;\" \/>. In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1df1320a6a7d764d3da05dd21adb50fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#99;&#100;&#111;&#116;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#53;&#92;&#100;&#105;&#118;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#53;&#92;&#100;&#105;&#118;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#53;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#49;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#53;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"702\" style=\"vertical-align: -16px;\" \/><\/p>\n<p id=\"fs-id1170654116562\">Division follows the same rules as multiplication!<\/p>\n<p id=\"fs-id1170653915002\">For division of two signed numbers, when the:<\/p>\n<ul id=\"fs-id1166426287557\" data-bullet-style=\"bullet\">\n<li>signs are the <em data-effect=\"italics\">same<\/em>, the quotient is <em data-effect=\"italics\">positive<\/em>.<\/li>\n<li>signs are <em data-effect=\"italics\">different<\/em>, the quotient is <em data-effect=\"italics\">negative<\/em>.<\/li>\n<\/ul>\n<p id=\"fs-id1170653869100\">And remember that we can always check the answer of a division problem by multiplying.<\/p>\n<div id=\"fs-id1170654015283\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Multiplication and Division of Signed Numbers<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p id=\"fs-id1170653774421\">For multiplication and division of two signed numbers:<\/p>\n<ul id=\"fs-id1166425927232\" data-bullet-style=\"bullet\">\n<li>If the signs are the same, the result is positive.<\/li>\n<li>If the signs are different, the result is negative.<\/li>\n<\/ul>\n<table id=\"fs-id1170654154997\" style=\"width: 100%;\" summary=\"No Summary\">\n<thead>\n<tr valign=\"top\">\n<th style=\"width: 50%;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Same signs<\/th>\n<th style=\"width: 50%;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Result<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"width: 50%;\" data-valign=\"top\" data-align=\"left\">Two positives<\/td>\n<td style=\"width: 50%;\" data-valign=\"top\" data-align=\"left\">Positive<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\">Two negatives<\/td>\n<td style=\"width: 50%;\">Positive<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>If the signs are the same, the result is positive.<\/p>\n<table id=\"fs-id1170653799921\" style=\"width: 100%; height: 48px;\" summary=\"No Summary\">\n<thead>\n<tr style=\"height: 16px;\" valign=\"top\">\n<th style=\"height: 16px; width: 50%;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Different signs<\/th>\n<th style=\"height: 16px; width: 50%;\" scope=\"col\" data-valign=\"top\" data-align=\"left\">Result<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 16px;\" valign=\"top\">\n<td style=\"height: 16px; width: 50%;\" data-valign=\"top\" data-align=\"left\">Positive and negative<\/td>\n<td style=\"height: 16px; width: 50%;\" data-valign=\"top\" data-align=\"left\">Negative<\/td>\n<\/tr>\n<tr style=\"height: 16px;\">\n<td style=\"height: 16px; width: 50%;\">Negative and positive<\/td>\n<td style=\"height: 16px; width: 50%;\">Negative<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>If the signs are different, the result is negative.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170652618826\" data-type=\"problem\">\n<p id=\"fs-id1170654008734\">Divide: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b3dbb76a7a3b4b23f8238a05c6710616_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#55;&#92;&#100;&#105;&#118;&#32;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3443fdd50bfd8cbef0209c257457ca50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;&#48;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654212229\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-762\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>a)<br \/>\nDivide. With different signs, the quotient is negative.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-abd0e0180314b3577a7a4ff8d744a8cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#50;&#55;&#92;&#100;&#105;&#118;&#32;&#51;&#92;&#92;&#32;&#45;&#57;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"61\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>b)<br \/>\nDivide. With signs that are the same, the quotient is positive.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1b9824277f2d9151cf02a1aaaf207d46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#49;&#48;&#48;&#92;&#100;&#105;&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#50;&#53;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"97\" style=\"vertical-align: -11px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170653797425\" data-type=\"problem\">\n<p id=\"fs-id1170653906270\">Divide: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d2160bf7a3c60d2097ea7ceab482e603_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#50;&#92;&#100;&#105;&#118;&#32;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e440c780a9ce4097c4c6094154a91c25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;&#55;&#92;&#100;&#105;&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654149901\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654047662\">a)<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-00c6d30c5f7439a21caf981437f64be1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/> b) 39<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Simplify Expressions with Integers<\/h1>\n<p id=\"fs-id1170653741550\">What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?<\/p>\n<p id=\"fs-id1170653790201\">Let\u2019s try some examples. We\u2019ll simplify expressions that use all four operations with integers\u2014addition, subtraction, multiplication, and division. Remember to follow the order of operations.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 13<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654066443\" data-type=\"problem\">\n<p id=\"fs-id1170653768124\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-24b8f0796e27dc6238f7631b9c932920_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170652623765\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-431\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-24b8f0796e27dc6238f7631b9c932920_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply first.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-44a4a8f04ead233888dbb3b890d6faf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-652e950e5cfb123bb379f94f5038aaef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#50;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-08ed8320699be2770837da731d65c7c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 13<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170652620764\" data-type=\"problem\">\n<p id=\"fs-id1170653875298\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bdf49d134a670670b6e654be9873bc8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654149857\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654149859\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c081a4d1588de9f9de1dd7d5282b4c9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170653871378\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170652630281\" data-type=\"exercise\">\n<div id=\"fs-id1170652620232\" data-type=\"solution\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 14<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170652619970\" data-type=\"problem\">\n<p id=\"fs-id1170653761875\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-39e546b1083a74884e4981f6405b861e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"42\" style=\"vertical-align: -4px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f278ff35c8dcfadef48d2edadefd1d3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#123;&#50;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"22\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654237484\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-217\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td>a)<br \/>\nWrite in expanded form.<br \/>\nMultiply.<br \/>\nMultiply.<br \/>\nMultiply.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-205c076c140d74f350b316871c06b34d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;&#92;&#92;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#45;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#49;&#54;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"153\" style=\"vertical-align: -46px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>b)<br \/>\nWrite in expanded form. We are asked to find the opposite of<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-71f34c438201a6a3b27229dd6727aefc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#50;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/>.<br \/>\nMultiply.<br \/>\nMultiply.<br \/>\nMultiply.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c0f9789031a3c497109479b1171effac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#123;&#50;&#125;&#94;&#123;&#52;&#125;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#99;&#100;&#111;&#116;&#50;&#92;&#99;&#100;&#111;&#116;&#50;&#92;&#99;&#100;&#111;&#116;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#99;&#100;&#111;&#116;&#50;&#92;&#99;&#100;&#111;&#116;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#92;&#99;&#100;&#111;&#116;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#49;&#54;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"103\" width=\"96\" style=\"vertical-align: -44px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1170653737864\">Notice the difference in parts a) and b). In part a), the exponent means to raise what is in the parentheses, the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5ef599d93da3e39727ed820e5302abcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/> to the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bc80ad7a002ec7cb8b7ec5deacae34ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#52;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"22\" style=\"vertical-align: -1px;\" \/> power. In part b), the exponent means to raise just the 2 to the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bc80ad7a002ec7cb8b7ec5deacae34ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#52;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"22\" style=\"vertical-align: -1px;\" \/> power and then take the opposite.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 14<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170652622523\" data-type=\"problem\">\n<p id=\"fs-id1170653790279\">Simplify: a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7f030f29170d3e23cfadee1ce2e2d79f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"42\" style=\"vertical-align: -4px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-91ffa9fdd7ca1075001d7087286a50bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#123;&#51;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"22\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654230619\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654230621\">a) 81 b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1e9ef1c6bd7a9b6e2b87026a76cda03b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170653904816\">The next example reminds us to simplify inside parentheses first.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 15<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170652622789\" data-type=\"problem\">\n<p id=\"fs-id1170652622791\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ec9bc12da033b8553035518caca5ec4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653905533\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-138\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ec9bc12da033b8553035518caca5ec4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Subtract in parentheses first.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-63d263cf0990efbc8163197da5164dc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"86\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fc7cc2bf045888e626ff99795f9cd64a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1b44aa122f73c22d86f2d764af01f919_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 15<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654134940\" data-type=\"problem\">\n<p id=\"fs-id1170654134942\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c4be2c0c69667c5427c3624021a9ade1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#55;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"112\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170652622244\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654150070\">29<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 16<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170652624014\" data-type=\"problem\">\n<p id=\"fs-id1170652624016\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3fd25bec589215019b11114b5a71de57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#100;&#105;&#118;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"113\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653850514\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<div data-type=\"title\"><\/div>\n<table id=\"eip-410\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3fd25bec589215019b11114b5a71de57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#100;&#105;&#118;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"113\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Exponents first.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0bcec5a1521eb6334fca6eb1035f845a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#100;&#105;&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"106\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-69278257962ca777f28ae178be1a5764_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#50;&#92;&#100;&#105;&#118;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e99b7c7ddd421682faf67e5b5f865882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 16<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170652621179\" data-type=\"problem\">\n<p id=\"fs-id1170652621181\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0711635a71c0d1ed162c9ada0c66dd3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#100;&#105;&#118;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"121\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170653797293\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170653797295\">4<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1170654156593\" class=\"try\" data-type=\"note\">\n<div id=\"fs-id1170652621176\" data-type=\"exercise\">\n<div id=\"fs-id1170652621179\" data-type=\"problem\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 17<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654047271\" data-type=\"problem\">\n<p id=\"fs-id1170654047273\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bcc4d59c21f76a152c670ba0d291b91c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#48;&#92;&#100;&#105;&#118;&#32;&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654233306\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"eip-343\" style=\"width: 100%;\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bcc4d59c21f76a152c670ba0d291b91c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#48;&#92;&#100;&#105;&#118;&#32;&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -5px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply and divide left to right, so divide first.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f745634152aacc8a73f83c4acbc56484_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-597a535a0f67f119498557f2d72a72c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;&#43;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"69\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b5c3e12330dabaeec7413281aba0f134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 17<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1170654154229\" data-type=\"problem\">\n<p id=\"fs-id1170654154231\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e736de0ffba98fa939559a4e1e80386b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#55;&#92;&#100;&#105;&#118;&#32;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<\/div>\n<div id=\"fs-id1170654079166\" data-type=\"solution\">\n<details>\n<summary>Show answer<\/summary>\n<p id=\"fs-id1170654079169\">21<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1170655091191\">Access these online resources for additional instruction and practice with adding and subtracting integers. You will need to enable Java in your web browser to use the applications.<\/p>\n<ul id=\"fs-id1170655096461\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/11AddColorChip\">Add Colored Chip<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/11SubtrColorChp\">Subtract Colored Chip<\/a><\/li>\n<\/ul>\n<\/div>\n<h1>Key Concepts<\/h1>\n<ul id=\"fs-id1170655097500\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Addition of Positive and Negative Integers<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-49e6bf8e8d878820068288f8323fc0bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#53;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#57;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#52;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#111;&#116;&#104;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#44;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#98;&#111;&#116;&#104;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#44;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#109;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#109;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#45;&#53;&#43;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#57;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#52;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#116;&#32;&#115;&#105;&#103;&#110;&#115;&#44;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#116;&#32;&#115;&#105;&#103;&#110;&#115;&#44;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#114;&#101;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#114;&#101;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#109;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#109;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"237\" width=\"320\" style=\"vertical-align: -113px;\" \/><\/li>\n<li><strong data-effect=\"bold\">Property of Absolute Value<\/strong>: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b6cd98f4266f7f491a976c3270ec303d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#110;&#124;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/> for all numbers. Absolute values are always greater than or equal to zero!<\/li>\n<li><strong data-effect=\"bold\">Subtraction of Integers<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-dc255b92a82dd3fd642e90f94d38cd9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#53;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#45;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#57;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#57;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#107;&#101;&#32;&#97;&#119;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#97;&#107;&#101;&#32;&#97;&#119;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#32;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#45;&#53;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#57;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#57;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#115;&#44;&#32;&#119;&#97;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#115;&#44;&#32;&#119;&#97;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#101;&#100;&#32;&#110;&#101;&#117;&#116;&#114;&#97;&#108;&#32;&#112;&#97;&#105;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#101;&#100;&#32;&#110;&#101;&#117;&#116;&#114;&#97;&#108;&#32;&#112;&#97;&#105;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"259\" width=\"372\" style=\"vertical-align: -124px;\" \/><\/li>\n<li><strong data-effect=\"bold\">Subtraction Property:<\/strong> Subtracting a number is the same as adding its opposite.<\/li>\n<li><strong data-effect=\"bold\">Multiplication and Division of Two Signed Numbers<\/strong>\n<ul id=\"fs-id1170653881237\" data-bullet-style=\"open-circle\">\n<li>Same signs\u2014Product is positive<\/li>\n<li>Different signs\u2014Product is negative<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h1>Glossary<\/h1>\n<div class=\"textbox shaded\">\n<dl id=\"fs-id1166424847478\">\n<dt>absolute value<\/dt>\n<dd id=\"fs-id1166424809754\">The absolute value of a number is its distance from 0 on the number line. The absolute value of a number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b38a586dc76a31dfa818cbaca2da26f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#110;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"17\" style=\"vertical-align: -4px;\" \/>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1166424772590\">\n<dt>integers<\/dt>\n<dd id=\"fs-id1166424772595\">The whole numbers and their opposites are called the integers: &#8230;\u22123, \u22122, \u22121, 0, 1, 2, 3&#8230;<\/dd>\n<\/dl>\n<dl id=\"fs-id1166422694257\">\n<dt>opposite<\/dt>\n<dd id=\"fs-id1166422694262\">The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-236972f3904fc6d64d7fafb16c1fe32d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"22\" style=\"vertical-align: 0px;\" \/> means the opposite of the number. The notation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-236972f3904fc6d64d7fafb16c1fe32d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"22\" style=\"vertical-align: 0px;\" \/> is read \u201cthe opposite of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>.\u201d<\/dd>\n<\/dl>\n<\/div>\n<h1>1.2 Exercise Set<\/h1>\n<p id=\"fs-id1166422643633\">In the following exercises, order each of the following pairs of numbers, using &lt; or &gt;.<\/p>\n<ol>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e99b7c7ddd421682faf67e5b5f865882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6111a899fd636b7a5238708f8679f6ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b5c3e12330dabaeec7413281aba0f134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" style=\"vertical-align: -1px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-24329a36a4e10288288979f77a565ca2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p id=\"fs-id1171791388513\">In the following exercises, simplify.<\/p>\n<ol start=\"2\">\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cf59ae45f118aabfd9dbcd53d172d47d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#51;&#50;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d3a2c72b2b6c1d226f102034f66b93ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#48;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"15\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-58539f6df9ac445c3f81bf3233b2dfe2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#49;&#54;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"24\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p id=\"fs-id1170655355498\">In the following exercises, fill in &lt;, &gt;, or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b8690b7efd237bfe32a6e92e3b699b96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"4\" width=\"13\" style=\"vertical-align: 2px;\" \/> for each of the following pairs of numbers.<\/p>\n<ol start=\"3\">\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6a85299f81969ab364296f0c764c3640_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#45;&#54;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3f8a38c993190bb32bb9fed507e2a202_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#124;&#45;&#51;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\" \/>___<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p id=\"fs-id1170655215530\">In the following exercises, simplify.<\/p>\n<ol class=\"twocolumn\" start=\"4\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-60683e15f2e6872938e897e32db83a14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#124;&#45;&#53;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"151\" style=\"vertical-align: -5px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a0698468bdb5d50245a3670bd550c597_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#124;&#45;&#55;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c025cb6f15bed386886aa63a6e7ab57a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#49;&#53;&#45;&#55;&#124;&#45;&#124;&#49;&#52;&#45;&#54;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7817b31b5d79341c225c1a320bddd44a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#56;&#45;&#124;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1166424899765\">In the following exercises, simplify each expression.<\/p>\n<ol class=\"twocolumn\" start=\"8\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-be75ab12ebe65505f17f14e762893b68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#49;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5347e78e1388867c78f84257e82214bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-070ef83fdc77ea566edbd4518f4c8c3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a41443d259e033deeaf23df0ae06bb8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#56;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-918e4de58e3bd439d356305182c48d8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#57;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#43;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1166422656200\">In the following exercises, simplify.<\/p>\n<ol class=\"twocolumn\" start=\"13\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0ce3eef413b98f0c4e456f424b41fff5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"39\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-02fcf1c91b817070cf9422bae84727ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"52\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c72327a70078ba8cd7a16901f1e23351_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2f6e73a75721e926ae49e3abb7b13a4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#52;&#45;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a2c343faa3d44761a7060b26e307afcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#52;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c731405c20b6850ef82dba43e0fc598b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c25edd25eccda1d968816153366e08f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#43;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: -2px;\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p id=\"fs-id1170655166333\">In the following exercises, simplify each expression.<\/p>\n<ol class=\"twocolumn\" start=\"18\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c927e3aec0bb8d8fbb7123a337953c86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cff4290119b5d26303e75458d7fa0f10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#45;&#56;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-edd48401ecbeced99097cef4de7a30e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#55;&#45;&#52;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"69\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-33bba2ed0a23d94f22907adacee12146_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;&#51;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4ff85d3fb26c79ccc9a42174b998e58a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7998a7664747147b3bf95fc27b9d3ff9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#45;&#51;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"70\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8c9845a10ae5cfef299283be86446d6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#45;&#52;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"83\" style=\"vertical-align: -2px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-18410c1f656e226b5e7a70abf2d5196c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#52;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e1d3f441ec8cd0340f54042a985dd6c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"152\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-99fbba525ec68f743df0b10075df01b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-93bff9c2baf995720da4030aee245f22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#091;&#49;&#48;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#093;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c0c067cb4ca65625221f9489630fb8c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#46;&#51;&#45;&#52;&#46;&#51;&#45;&#55;&#46;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"111\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fe86fd15a2527313d188aeee5053560b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#53;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#54;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"54\" style=\"vertical-align: 0px;\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1166421558397\">In the following exercises, multiply.<\/p>\n<ol class=\"twocolumn\" start=\"31\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6c35d4f5986bc5f7c4127a38b882fe2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#92;&#99;&#100;&#111;&#116;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-02d9f5a32b08eb8d149cd6577bcb3d70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#92;&#99;&#100;&#111;&#116;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b361129d452a3d90c0bec2bb004052be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7d398b90f253a04cd8d61419909697cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1166421949046\">In the following exercises, divide.<\/p>\n<ol class=\"twocolumn\" start=\"35\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5bf127a98d89e04d3ddc7f28746db24e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#52;&#92;&#100;&#105;&#118;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cba9138f78ae5ea833c8990a4b3ac841_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#56;&#48;&#92;&#100;&#105;&#118;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"78\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-633a865ee9b7d2b894c6b3e808aa8369_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#50;&#92;&#100;&#105;&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -5px;\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1166421486016\">In the following exercises, simplify each expression.<\/p>\n<ol class=\"twocolumn\" start=\"38\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-85f0f85521a080391f767cabf8d51180_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c3d91d19a200676911902f50f3ee4e43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"42\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6a686631829c9f458dd8dc32b0614e8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#123;&#52;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-aa75b0df6dbc1e81f7fd2a16258eb2f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"86\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e0394b023c4f67d69d7a99987647f7f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-653f4ea6625efc3317803561a1dcf962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#54;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0b6d57327d04caa90d36199cca298b0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#53;&#92;&#100;&#105;&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#100;&#105;&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"200\" style=\"vertical-align: -5px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d53067a8da43befde6f153cea9ad046d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#091;&#51;&#45;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#093;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6548c84526f3f5182cb18e9dbea8ff39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#92;&#100;&#105;&#118;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"157\" style=\"vertical-align: -5px;\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1166423907674\">In the following exercises, solve.<\/p>\n<ol start=\"47\">\n<li><strong data-effect=\"bold\">Temperature<\/strong> On January <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b59223d2c162a0fdd163af8597f0845e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/>, the high temperature in Lytton, British Columbia, was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2044c4aefe2adfc3b277146852400fd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: -1px;\" \/>\u00b0 . That same day, the high temperature in Fort Nelson, British Columbia was <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9eae543c53f9301fa411ef541a1f4433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/>\u00b0. What was the difference between the temperature in Lytton and the temperature in Embarrass?<\/li>\n<li><strong data-effect=\"bold\"> Checking Account<\/strong> Ester has &#36;124 in her checking account. She writes a check for &#36;152. What is the new balance in her checking account?<\/li>\n<li><strong data-effect=\"bold\">Checking Account<\/strong> Kevin has a balance of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c09bd1dab7b4ad4642548bdff771c81a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#36;&#51;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -1px;\" \/> in his checking account. He deposits &#36;225 to the account. What is the new balance?<\/li>\n<li>\n<p id=\"fs-id1170654920103\"><strong>Provincial budgets<\/strong> For 2019 the province of Quebec estimated it would have a budget surplus of &#36;5.6 million. That same year, Alberta estimated it would have a budget deficit of &#36;7.5 million.<\/p>\n<p id=\"fs-id1170655202258\">Use integers to write the budget of:<\/p>\n<ol type=\"a\">\n<li>Quebec<\/li>\n<li>Alberta<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<h1>Answers:<\/h1>\n<ol class=\"threecolumn\">\n<li>\n<ol type=\"a\">\n<li>&gt;<\/li>\n<li>&lt;<\/li>\n<li>&lt;<\/li>\n<li>&gt;<\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li>32<\/li>\n<li>0<\/li>\n<li>16<\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li>&lt;<\/li>\n<li>=<\/li>\n<\/ol>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5a46791a391dd60ec10b6fdd529ca8ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#44;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"39\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>56<\/li>\n<li>0<\/li>\n<li>8<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3f7920c1ff891a318599e38fe5bcd7ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/li>\n<li>32<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-851674cf8d214f2dc493930ee7e6ee73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/><\/li>\n<li>108<\/li>\n<li>29<\/li>\n<li>6<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/li>\n<li>12<\/li>\n<li>\n<ol type=\"a\">\n<li>16<\/li>\n<li>16<\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"a\">\n<li>45<\/li>\n<li>45<\/li>\n<\/ol>\n<\/li>\n<li>27<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bf64b03624682b0a337c10836b136ca1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c8592fdae394223d029fcc21317133ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-20a7f6e08b5f463553128808e5b96339_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/li>\n<li>-99<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/li>\n<li>22<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0329d19b932005c996d83dbb9e09ffed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/li>\n<li>4<\/li>\n<li>6<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-65f7efd0fbf95ed3f6862d4633b9b62b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#46;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2981688f5b26fd15fa08c897afa741ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-fce2979511e6a27a1bd61ca970c655b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c081a4d1588de9f9de1dd7d5282b4c9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/li>\n<li>14<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9eae543c53f9301fa411ef541a1f4433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/li>\n<li>13<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6193f2be927539d70e972d662785fd1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/li>\n<li>64<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-60f20f07f8f7c62331f2d8fe388a61b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/li>\n<li>90<\/li>\n<li>9<\/li>\n<li>41<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9672181aec15f8334b80ada7de4e4fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-94535250d2f2556f9c0cb93ff04db6e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/li>\n<li>5<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-662f6ae8c8d225801ecf7f82bc73ef62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/>\u00b0<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-2908b59f3cefca439cfc06f4a261b53f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#36;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -1px;\" \/><\/li>\n<li>&#36;187<\/li>\n<li>\n<ol type=\"a\">\n<li>&#36;5.6 million<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b639daa2e3da1c3138a62bb15a983679_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#36;&#55;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"44\" style=\"vertical-align: -1px;\" \/> million<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div data-type=\"glossary\">\n<h1>Attributions<\/h1>\n<p>This chapter has been adapted from \u201cAdd and Subtract Integers\u201d in <a href=\"https:\/\/openstax.org\/details\/books\/elementary-algebra\"><em>Elementary Algebra<\/em><\/a> (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 Licence<\/a>. Adapted by Izabela Mazur. See the Copyright page for more information.<\/p>\n<\/div>\n","protected":false},"author":125,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-120","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/120","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/users\/125"}],"version-history":[{"count":3,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/120\/revisions"}],"predecessor-version":[{"id":2150,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/120\/revisions\/2150"}],"part":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/120\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/media?parent=120"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapter-type?post=120"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/contributor?post=120"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/license?post=120"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}