{"id":1438,"date":"2019-07-29T23:25:31","date_gmt":"2019-07-29T23:25:31","guid":{"rendered":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/graphing-systems-of-linear-inequalities\/"},"modified":"2021-08-31T21:21:22","modified_gmt":"2021-08-31T21:21:22","slug":"graphing-systems-of-linear-inequalities","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/businesstechnicalmath\/chapter\/graphing-systems-of-linear-inequalities\/","title":{"raw":"4.5 Graphing Systems of Linear Inequalities","rendered":"4.5 Graphing Systems of Linear Inequalities"},"content":{"raw":"[latexpage]\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nBy the end of this section it is expected that you will be able to:\n<ul>\n \t<li>Determine whether an ordered pair is a solution of a system of linear inequalities<\/li>\n \t<li>Solve a system of linear inequalities by graphing<\/li>\n \t<li>Solve applications of systems of inequalities<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345636018\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities<\/h1>\n<p id=\"fs-id1168344338607\">The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.<\/p>\n\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">System of Linear Inequalities<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nTwo or more linear inequalities grouped together form a <span data-type=\"term\">system of linear inequalities<\/span>.\n\n<\/div>\n<\/div>\n<p id=\"fs-id1168345550211\">A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown below.<\/p>\n\n<div id=\"fs-id1168344523603\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(\\left\\{\\begin{array}{c}x+4y\\ge 10\\hfill \\\\ 3x-2y&lt;12\\hfill \\end{array}\\)<\/div>\n<p id=\"fs-id1168341972977\">To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs \\(\\left(x,y\\right)\\) that make both inequalities true.<\/p>\n\n<div data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Solutions of a System of Linear Inequalities<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nSolutions of a system of linear inequalities are the values of the variables that make all the inequalities true.\n\n<\/div>\n<\/div>\n<span style=\"text-align: initial; font-size: 14pt;\">The solution of a system of linear inequalities is shown as a shaded region in the <\/span><em style=\"text-align: initial; font-size: 14pt;\" data-effect=\"italics\">x-y<\/em><span style=\"text-align: initial; font-size: 14pt;\"> coordinate system that includes all the points whose ordered pairs make the inequalities true.<\/span>\n\n<\/div>\n<\/div>\nTo determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1168345198719\">Determine whether the ordered pair is a solution to the system. \\(\\left\\{\\begin{array}{c}x+4y\\ge 10\\hfill \\\\ 3x-2y&lt;12\\hfill \\end{array}\\)<\/p>\n<p id=\"fs-id1168345397911\"><span class=\"token\">a)<\/span> (\u22122, 4) b) (3,1)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345440026\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n\na) Is the ordered pair (\u22122, 4) a solution?<span data-type=\"newline\">\n<\/span><span id=\"fs-id1168345428380\" data-type=\"media\" data-alt=\"This figure says, \u201cWe substitute x = -2 and y = 4 into both inequalities. The first inequality, x + 4 y is greater than or equal to 10 becomes -2 plus 4 times 4 is greater than or less than 10 or 14 is great than or less than 10 which is true. The second inequality, 3x \u2013 2y is less than 12 becomes 3 times -2 \u2013 2 times 4 is less than 12 or -14 is less than 12 which is true.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2019\/07\/CNX_ElemAlg_Figure_05_06_001_img_new.jpg\" alt=\"This figure says, \u201cWe substitute x = -2 and y = 4 into both inequalities. The first inequality, x + 4 y is greater than or equal to 10 becomes -2 plus 4 times 4 is greater than or less than 10 or 14 is great than or less than 10 which is true. The second inequality, 3x \u2013 2y is less than 12 becomes 3 times -2 \u2013 2 times 4 is less than 12 or -14 is less than 12 which is true.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1167834065902\">The ordered pair (\u22122, 4) made both inequalities true. Therefore (\u22122, 4) is a solution to this system.<\/p>\nb) Is the ordered pair (3,1) a solution?<span data-type=\"newline\">\n<\/span><span id=\"fs-id1168345417372\" data-type=\"media\" data-alt=\"This figure says, \u201cWe substitute x 3 and y = 1 into both inequalities.\u201d The first inequality, x + 4y is greater than or equal to 10 becomes 3 + 4 times 1 is greater than or equal to 10 or y is greater than or equal to 10 which is false. The second inequality, 3x -2y is less than 12 becomes 3 times 3 \u2013 two times 1 is less than 12 or 7 is less than 12 which is true.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_002_img_new.jpg\" alt=\"This figure says, \u201cWe substitute x 3 and y = 1 into both inequalities.\u201d The first inequality, x + 4y is greater than or equal to 10 becomes 3 + 4 times 1 is greater than or equal to 10 or y is greater than or equal to 10 which is false. The second inequality, 3x -2y is less than 12 becomes 3 times 3 \u2013 two times 1 is less than 12 or 7 is less than 12 which is true.\" data-media-type=\"image\/jpeg\"><\/span>\n<p id=\"fs-id1168345327228\">The ordered pair (3,1) made one inequality true, but the other one false. Therefore (3,1) is not a solution to this system.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 1<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n\nDetermine whether the ordered pair is a solution to the system.\n<div id=\"fs-id1168345253745\" data-type=\"note\">\n<div id=\"fs-id1168345510038\" data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1168345191853\"><img class=\"alignnone wp-image-1381\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.07.43-PM-300x120.png\" alt=\"\" width=\"140\" height=\"56\"><\/p>\n<p id=\"fs-id1168345436310\">a) \\(\\left(3,-1\\right)\\) b) \\(\\left(6,-3\\right)\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345556267\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345745212\">a) no b) yes<\/p>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve a System of Linear Inequalities by Graphing<\/h1>\n<p id=\"fs-id1168345415085\">The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"title\">How to Solve a System of Linear inequalities<\/div>\n<div id=\"fs-id1168345385725\" data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1168345384150\">Solve the system by graphing.<\/p>\n\\(\\left\\{\\begin{array}{c}y\\ge 2x-1\\hfill \\\\ y&lt;x+1\\hfill \\end{array}\\)\n\n<\/div>\n<div 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<span id=\"fs-id1168345191155\" data-type=\"media\" data-alt=\"This is a table with three columns and several rows. The first row says, \u201cStep 1: Graph the first inequality. We will graph y is greater than or equal to 2x \u2013 1.\u201d There are two equations givens, y is greater than or equal to 2x \u2013 1 and y is less than x + 1. The table then reads, \u201cGraph the boundary line. We graph the line y = 2x \u2013 1. It is a solid line because the inequality sign is greater than or equal to. Shade in the side of the boundary line where the inequality is true. We choose (0, 0) as a test point. It is a solution to y is greater than or equal to 2x \u2013 1, so we shad in the left side of the boundary line.\u201d There is a figure of a line graphed on an x y coordinate plane. The area to the left of the line is shaded.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_003a_img_new.jpg\" alt=\"This is a table with three columns and several rows. The first row says, \u201cStep 1: Graph the first inequality. We will graph y is greater than or equal to 2x \u2013 1.\u201d There are two equations givens, y is greater than or equal to 2x \u2013 1 and y is less than x + 1. The table then reads, \u201cGraph the boundary line. We graph the line y = 2x \u2013 1. It is a solid line because the inequality sign is greater than or equal to. Shade in the side of the boundary line where the inequality is true. We choose (0, 0) as a test point. It is a solution to y is greater than or equal to 2x \u2013 1, so we shad in the left side of the boundary line.\u201d There is a figure of a line graphed on an x y coordinate plane. The area to the left of the line is shaded.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345409450\" data-type=\"media\" data-alt=\"The second row then says, \u201cStep 2: On the same grid, graph the second inequality. We will graph y is less than x + 1 on the same grid. Grph the boundary line. We graph the lin y = x + 1. It is a dashed line because the inequality sign is less than. There is a graph which shows two lines graphed on an x y coordinate plane. The area to the left of one line is shaded. The area to the right of the second line is shaded. There is a small area where the shaded areas overlap. The table then says, \u201cShade in the side of that boundary line where the inequality is true. Again we use (0, 0) as a test point. It is a solution so we shade in that side of the line y = x + 1.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_003b_img_new.jpg\" alt=\"The second row then says, \u201cStep 2: On the same grid, graph the second inequality. We will graph y is less than x + 1 on the same grid. Grph the boundary line. We graph the lin y = x + 1. It is a dashed line because the inequality sign is less than. There is a graph which shows two lines graphed on an x y coordinate plane. The area to the left of one line is shaded. The area to the right of the second line is shaded. There is a small area where the shaded areas overlap. The table then says, \u201cShade in the side of that boundary line where the inequality is true. Again we use (0, 0) as a test point. It is a solution so we shade in that side of the line y = x + 1.\" data-media-type=\"image\/jpeg\"><\/span><span id=\"fs-id1168345209236\" data-type=\"media\" data-alt=\"The third row then says, \u201cStep 3: The solution is the region where the shading overlaps. The poing where the boundary lines intersect is not a solution because it is not a solution to y is less than x + 1. The solution is all points in the purple shaded region.\u201d\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_003c_img_new.jpg\" alt=\"The third row then says, \u201cStep 3: The solution is the region where the shading overlaps. The poing where the boundary lines intersect is not a solution because it is not a solution to y is less than x + 1. The solution is all points in the purple shaded region.\u201d\" data-media-type=\"image\/jpeg\"><\/span><span data-type=\"media\" data-alt=\"The fourth row then says, \u201cStep 4: Check by choosing a test point. We\u2019ll use (-1, -1) as a test point. Is (-1, -1) a solution to y is greater than or equal to 2x \u2013 1? -1 is greater than or equal to 2 times -1 \u2013 1 or -1 is greater than or equal to -3 true.\u201d\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_003d_img_new.jpg\" alt=\"The fourth row then says, \u201cStep 4: Check by choosing a test point. We\u2019ll use (-1, -1) as a test point. Is (-1, -1) a solution to y is greater than or equal to 2x \u2013 1? -1 is greater than or equal to 2 times -1 \u2013 1 or -1 is greater than or equal to -3 true.\u201d\" data-media-type=\"image\/jpeg\"><\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 2<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n\nSolve the system by graphing. <img class=\"alignnone wp-image-1386\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-try-it-2-300x134.png\" alt=\"\" width=\"141\" height=\"63\">\n\n<\/div>\n<div id=\"fs-id1168345374077\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary><span id=\"fs-id1168345455090\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is less than 3x +2 and y is greater than \u2013x \u2013 1. The area to the right of each line is shaded slightly different colors with the overlapping area also shaded a slightly different color. Both lines are dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_013_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is less than 3x +2 and y is greater than \u2013x \u2013 1. The area to the right of each line is shaded slightly different colors with the overlapping area also shaded a slightly different color. Both lines are dotted.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\"><span style=\"font-size: 14pt;\">Solve a system of linear inequalities by graphing.<\/span><\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1169753926787\" class=\"stepwise\" type=\"1\">\n \t<li>Graph the first inequality.\n<ul id=\"fs-id1168345277927\" data-bullet-style=\"bullet\">\n \t<li>Graph the boundary line.<\/li>\n \t<li>Shade in the side of the boundary line where the inequality is true.<\/li>\n<\/ul>\n<\/li>\n \t<li>On the same grid, graph the second inequality.\n<ul id=\"fs-id1168345399311\" data-bullet-style=\"bullet\">\n \t<li>Graph the boundary line.<\/li>\n \t<li>Shade in the side of that boundary line where the inequality is true.<\/li>\n<\/ul>\n<\/li>\n \t<li>The solution is the region where the shading overlaps.<\/li>\n \t<li>Check by choosing a test point.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"howto\" data-type=\"note\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 3<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168341961505\" data-type=\"problem\">\n\nSolve the system by graphing. <img class=\"alignnone wp-image-1388\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.11.42-PM-300x154.png\" alt=\"\" width=\"140\" height=\"72\">\n\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167835369328\" style=\"width: 100%;\" summary=\"This figure has both words and a graph. The words say, \u201cGraph x \u2013 y is greater than 3, by graphing x \u2013 y = 3 and testing a point. The intercepts are x = 3 and y = -3 and the boundary line will be dashed. Test (0, 0) which makes the inequality false so shade (red) the side that does not contain (0, 0).\u201d The graph shows a line graphed on an x y-coordinate plane. The line is dashed and the area to the right of the line is shaded.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 58.4091%;\" data-valign=\"top\">Graph\u00a0<em data-effect=\"italics\">x<\/em>\u00a0\u2212\u00a0<em data-effect=\"italics\">y<\/em>\u00a0&gt; 3, by graphing\u00a0<em data-effect=\"italics\">x<\/em>\u00a0\u2212\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= 3 and<span data-type=\"newline\">\n<\/span>testing a point.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>The intercepts are\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 3 and\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= \u22123 and the boundary<span data-type=\"newline\">\n<\/span>line will be dashed.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Test (0, 0). It makes the inequality false. So,<span data-type=\"newline\">\n<\/span>shade the side that does not contain (0, 0) red.<\/td>\n<td style=\"width: 41.4773%;\" data-valign=\"top\"><span id=\"fs-id1167831213870\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_004a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span>\n\n<span id=\"fs-id1167831213870\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_004a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 58.4091%;\" data-valign=\"top\">Graph \\(y&lt;-\\frac{1}{5}x+4\\) by graphing \\(y=-\\frac{1}{5}x+4\\)<span data-type=\"newline\">\n<\/span>using the slope \\(m=-\\frac{1}{5}\\) and\u00a0<em data-effect=\"italics\">y<\/em>\u2212intercept<span data-type=\"newline\">\n<\/span><em data-effect=\"italics\">b<\/em>\u00a0= 4. The boundary line will be dashed.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Test (0, 0). It makes the inequality true, so shade the side that contains (0, 0) blue.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Choose a test point in the solution and verify that it is a solution to both inequalities.<\/td>\n<td style=\"width: 41.4773%;\"><span id=\"fs-id1167831213870\" data-type=\"media\" data-alt=\".\"><span id=\"fs-id1167835369155\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_005a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nThe point of intersection of the two lines is not included as both boundary lines were dashed. The solution is the area shaded twice which is the darker-shaded region.\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 3<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"note\">\n\nSolve the system by graphing. \\(\\left\\{\\begin{array}{c}x+y\\le 2\\hfill \\\\ y\\ge \\frac{2}{3}x-1\\hfill \\end{array}\\)\n\n<\/div>\n<div id=\"fs-id1168341906364\" data-type=\"note\">\n<div data-type=\"exercise\">\n<div id=\"fs-id1168345260430\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary><span id=\"fs-id1168345436372\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of x + y is less than or equal to 2 and y is greater than or equal to (2\/3)x \u2013 1. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_027_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of x + y is less than or equal to 2 and y is greater than or equal to (2\/3)x \u2013 1. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"solution\">\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 4<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345337187\" data-type=\"problem\">\n<p id=\"fs-id1168345385902\">Solve the system by graphing. <img class=\"alignnone wp-image-1392\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.14.43-PM-300x147.png\" alt=\"\" width=\"141\" height=\"69\"><\/p>\n\n<\/div>\n<div id=\"fs-id1168341864081\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167835357015\" class=\"unnumbered unstyled can-break\" summary=\"This figure has both words and a graph. The words say, \u201cGraph x \u2013 2y is less than 5, by graphing x \u2013 2y = 5 and testing a point. The intercepts are x = 5 and y = -2.5 and the boundary line will be dashed. Test (0, 0) which makes the inequality true, so shade (red) the side that contains (0, 0).\u201d The figure shows a dashed line graphed on an x y-coordinate plane. The area above the line is shaded.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">Graph \\(x-2y&lt;5\\), by graphing \\(x-2y=5\\) and testing a point.<span data-type=\"newline\">\n<\/span>The intercepts are\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 5 and\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= \u22122.5 and the boundary line will be dashed.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Test (0, 0). It makes the inequality true. So, shade the side<span data-type=\"newline\">\n<\/span>that contains (0, 0) red.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167831911196\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_006a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>Graph\u00a0<em data-effect=\"italics\">y<\/em>\u00a0&gt; \u22124, by graphing\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= \u22124 and recognizing that it is a<span data-type=\"newline\">\n<\/span>horizontal line through\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= \u22124. The boundary line will be dashed.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Test (0, 0). It makes the inequality true. So, shade (blue)<span data-type=\"newline\">\n<\/span>the side that contains (0, 0) blue.<\/td>\n<td><span id=\"fs-id1167831911196\" data-type=\"media\" data-alt=\".\"><span id=\"fs-id1167831896589\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_007a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345255474\">The point (0, 0) is in the solution and we have already found it to be a solution of each inequality. The point of intersection of the two lines is not included as both boundary lines were dashed.<\/p>\nThe solution is the area shaded twice which is the darker-shaded region.\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 4<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1168345325877\">Solve the system by graphing. \\(\\left\\{\\begin{array}{c}y\\ge 3x-2\\hfill \\\\ y&lt;-1\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div id=\"fs-id1168345390829\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_028_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to 3x - 2 and y is less than -1. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\" data-media-type=\"image\/jpeg\">\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<span style=\"text-align: initial; font-size: 14pt;\">Systems of linear inequalities where the boundary lines are parallel might have no solution. We\u2019ll see this in the next example<\/span><span style=\"text-align: initial; font-size: 14pt;\">.<\/span>\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 5<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345241928\" data-type=\"problem\">\n<p id=\"fs-id1168345743048\">Solve the system by graphing. \\(\\left\\{\\begin{array}{c}4x+3y\\ge 12\\hfill \\\\ y&lt;-\\frac{4}{3}x+1\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167834279222\" class=\"unnumbered unstyled can-break\" summary=\"This figure has both words and a graph. The first set of words says, \u201cGraph 4x + 3y is greater than or equal to 12, by graphing 4x + 3 = 12 and testing a point. The intercepts are x = 3 and y = 4 and the boundary line will be solid. Test (0, 0) which makes the inequality false, s oshade (blue) the side that does not contain (0, 0).\u201d The figure shows a line graphed on an x y-coordinate plane. The line is solid, and the area to the right of the line is shaded. The second set of words says, \u201cGraph y is less than (-4\/3)x + 1 by graphing y = (-4\/3)x + 1 using the slope m = (-4\/3) and the y-intercept b = 1. The boundarly line will be dashed. Test (0, 0) which makes the inequality true, so shade (red) the side that contains (0, 0). The figure shows two line graphed on an x y-coordinate plane. One line is solid, and the area to the right is shaded. The other line is dashed, and the area to the left is shaded. There is no area of overlap between the two shaded areas.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">Graph \\(4x+3y\\ge 12\\), by graphing \\(4x+3y=12\\) and testing a point.<span data-type=\"newline\">\n<\/span>The intercepts are\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 3 and\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= 4 and the boundary line will be solid.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Test (0, 0). It makes the inequality false. So,<span data-type=\"newline\">\n<\/span>shade the side that does not contain (0, 0) red.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167834423065\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_008a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Graph \\(y&lt;-\\frac{4}{3}x+1\\) by graphing \\(y=-\\frac{4}{3}x+1\\) using the<span data-type=\"newline\">\n<\/span>slope \\(m=\\frac{4}{3}\\) and the\u00a0<em data-effect=\"italics\">y<\/em>-intercept\u00a0<em data-effect=\"italics\">b<\/em>\u00a0= 1. The boundary line will be dashed.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Test (0, 0). It makes the inequality true. So,<span data-type=\"newline\">\n<\/span>shade the side that contains (0, 0) blue.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167831893018\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_008b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345360332\">There is no point in both shaded regions, so the system has no solution. This system has no solution.<\/p>\n\n<\/div>\n&nbsp;\n\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 5<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345190544\" data-type=\"problem\">\n<p id=\"fs-id1168345407920\">Solve the system by graphing. \\(\\left\\{\\begin{array}{c}3x-2y\\le 12\\hfill \\\\ y\\ge \\frac{3}{2}x+1\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345260108\">no solution<\/p>\n<img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_029_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 3x \u2013 2y is less than or equal 12 and y is greater than or equal to (3\/2)x + 1. The area to the left or right of each line is shaded different colors. There is not overlapping area.\" data-media-type=\"image\/jpeg\">\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 6<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345230029\" data-type=\"problem\">\n\nSolve the system by graphing. <img class=\"alignnone wp-image-1399\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.16.28-PM-300x125.png\" alt=\"\" width=\"142\" height=\"59\">\n\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167835330830\" style=\"width: 100%;\" summary=\"This figure has both words and a graph. The first set of words says, \u201cGraph y is greater than (1\/2)x \u2013 4 by graphing y = (1\/2)x \u2013 4 using the slope m = 1\/2 and the intercept b = -4. The boundary line will be dashed. Tess (0, 0) which makes the inequality true, so shade (red) the side that contains (0, 0).\u201d The graph shows a dashed line graphed on an x y-coordinate plane with the area above it shaded. The second set of words says, \u201cGraph x -2 y is less than -4, by graphing x \u2013 2y = -4 and testing a point. The intercepts are x = -4 and y = 2 and the boundary line will be dashed. Choose a test point in the solution and verify that it is a solution to both inequalities.\u201d The graph shows to lines graphed. One is dashed, and the area above it is shaded. The other line is solid, and the area above it is shaded. There is an area where the shaded areas overlap.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">Graph \\(y\\) &gt; \\(\\frac{1}{2}x-4\\) by graphing \\(y=\\frac{1}{2}x-4\\)<span data-type=\"newline\">\n<\/span>using the slope \\(m=\\frac{1}{2}\\) and the intercept<span data-type=\"newline\">\n<\/span><em data-effect=\"italics\">b<\/em>\u00a0= \u22124. The boundary line will be dashed.<span data-type=\"newline\">\n<\/span>Test (0, 0). It makes the inequality true. So,<span data-type=\"newline\">\n<\/span>shade the side that contains (0, 0) red.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167834110177\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_009a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Graph \\(x-2y&lt;-4\\) by graphing \\(x-2y=-4\\) and testing a point.<span data-type=\"newline\">\n<\/span>The intercepts are\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= \u22124 and\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= 2 and the boundary<span data-type=\"newline\">\n<\/span>line will be dashed.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>Choose a test point in the solution and verify<span data-type=\"newline\">\n<\/span>that it is a solution to both inequalities.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167835309321\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_009b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345458774\">No point on the boundary lines is included in the solution as both lines are dashed.<\/p>\n<p id=\"fs-id1168345284829\">The solution is the region that is shaded twice, which is also the solution to \\(x-2y&lt;-4\\).<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 6<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345241699\" data-type=\"problem\">\n\nSolve the system by graphing. \\(\\left\\{\\begin{array}{c}y\\ge 3x+1\\hfill \\\\ -3x+y\\ge -4\\hfill \\end{array}\\)\n\n<\/div>\n<div id=\"fs-id1168345443262\" data-type=\"solution\"><details><summary class=\"answer\">Show answer<\/summary>\\(y\\ge 3x+1\\)\n\n<span id=\"fs-id1168345270107\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to 3x + 1 and -3x + y is greater than or equal to -4. The area to the left of each line is shaded with the overlapping area shaded a slightly different color.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_021_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to 3x + 1 and -3x + y is greater than or equal to -4. The area to the left of each line is shaded with the overlapping area shaded a slightly different color.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345251566\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve Applications of Systems of Inequalities<\/h1>\n<p id=\"fs-id1168345560133\">The first thing we\u2019ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system as we did above to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so their graphs will only show Quadrant I.<\/p>\n\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 7<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345273545\" data-type=\"problem\">\n<p id=\"fs-id1168345384897\">Christy sells her photographs at a booth at a street fair. At the start of the day, she wants to have at least 25 photos to display at her booth. Each small photo she displays costs her ?4 and each large photo costs her ?10. She doesn\u2019t want to spend more than ?200 on photos to display.<\/p>\n<p id=\"fs-id1169749520125\"><span class=\"token\">a)<\/span>\u00a0Write a system of inequalities to model this situation.<\/p>\n<p id=\"fs-id1169754128995\"><span class=\"token\">b)<\/span>\u00a0Graph the system.<\/p>\n<p id=\"fs-id1169752186836\"><span class=\"token\">c)<\/span>\u00a0Could she display 15 small and 5 large photos?<\/p>\n<p id=\"fs-id1169754046303\"><span class=\"token\">d)<\/span>\u00a0Could she display 3 large and 22 small photos?<\/p>\n\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n&nbsp;\n\na) Let \\(x=\\) the number of small photos.<span data-type=\"newline\">\n<\/span>\\(\\phantom{\\rule{2.5em}{0ex}}y=\\) the number of large photos<span data-type=\"newline\">\n<\/span>To find the system of inequalities, translate the information.<span data-type=\"newline\">\n<\/span>\\(\\begin{array}{ccccc}&amp; &amp; &amp; &amp; \\text{She wants to have at least 25 photos.}\\hfill \\\\ &amp; &amp; &amp; &amp; \\text{The number of small plus the number of large should be at least 25.}\\hfill \\\\ &amp; &amp; &amp; &amp; \\hfill x+y\\ge 25\\hfill \\\\ &amp; &amp; &amp; &amp; \\text{\\$4 for each small and \\$10 for each large must be no more than \\$200}\\hfill \\\\ &amp; &amp; &amp; &amp; \\hfill 4x+10y\\le 200\\hfill \\end{array}\\)<span data-type=\"newline\">\n<\/span>We have our system of inequalities. \\(\\left\\{\\begin{array}{c}x+y\\ge 25\\hfill \\\\ 4x+10y\\le 200\\hfill \\end{array}\\)\n\nb)\n<table id=\"fs-id1167834387646\" class=\"unnumbered unstyled can-break\" summary=\"This figure has words and a graph. The words say, \u201cTo graph x + y is greater than or equal 25, graph x + y = 25 as a solid line. Choose (0, 0) as a test point. Since it does not make the inequality true, shade (red) the side that does not include the point (0, 0). To graph 4x + 10y is less than or equal to 200, graph 4x + 10y = 200 as a solid line. Choose (0, 0) as a test point. Since it does make the inequality true, shade (blue) the side that include the point (0, 0). The graph show two line graphed on an x y-coordinte plane. One line has the area to the right shaded. The other line has the area to the left shaded. There is an area where the shaded areas overlap.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">To graph \\(x+y\\ge 25\\), graph\u00a0<em data-effect=\"italics\">x<\/em>\u00a0+\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= 25 as a solid line.<span data-type=\"newline\">\n<\/span>Choose (0, 0) as a test point. Since it does not make the inequality<span data-type=\"newline\">\n<\/span>true, shade the side that does not include the point (0, 0) red.<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>To graph \\(4x+10y\\le 200\\), graph 4<em data-effect=\"italics\">x<\/em>\u00a0+ 10<em data-effect=\"italics\">y<\/em>\u00a0= 200 as a solid line.<span data-type=\"newline\">\n<\/span>Choose (0, 0) as a test point. Since it does not make the inequality<span data-type=\"newline\">\n<\/span>true, shade the side that includes the point (0, 0) blue.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167835326332\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_010a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span data-type=\"newline\">\n<\/span>The solution of the system is the region of the graph that is double shaded and so is shaded darker.\n\nc) To determine if 10 small and 20 large photos would work, we see if the point (10, 20) is in the solution region. It is not. Christy would not display 10 small and 20 large photos.\n\nd) To determine if 20 small and 10 large photos would work, we see if the point (20, 10) is in the solution region. It is. Christy could choose to display 20 small and 10 large photos.\n<p id=\"fs-id1168345418460\">Notice that we could also test the possible solutions by substituting the values into each inequality.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 7<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345329549\" data-type=\"note\">\n<div id=\"fs-id1168345269797\" data-type=\"exercise\">\n<div id=\"fs-id1168345406421\" data-type=\"problem\">\n\nMary needs to purchase supplies of answer sheets and pencils for a standardized test to be given to the juniors at her high school. The number of the answer sheets needed is at least 5 more than the number of pencils. The pencils cost \\$2 and the answer sheets cost \\$1. Mary\u2019s budget for these supplies allows for a maximum cost of \\$400.\n<p id=\"fs-id1163872103544\">a) Write a system of inequalities to model this situation.\nb) Graph the system.\nc) Could Mary purchase 100 pencils and 100 answer sheets?\nd) Could Mary purchase 150 pencils and 150 answer sheets?<\/p>\n\n<\/div>\n<div id=\"fs-id1168345291918\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>a) \\(\\left\\{\\begin{array}{c}a\\ge p+5\\hfill \\\\ a+2p\\le 400\\hfill \\end{array}\\)\n\nb)\n<span id=\"fs-id1168345325509\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of a is greater than or equal to p + 5 and a + 2p is less than or equal to 400. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_030_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of a is greater than or equal to p + 5 and a + 2p is less than or equal to 400. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\"><\/span>\n\nc) no\n\nd) no\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">EXAMPLE 8<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345447989\" data-type=\"problem\">\n<p id=\"fs-id1168341862306\">Omar needs to eat at least 800 calories before going to his team practice. All he wants is hamburgers and cookies, and he doesn\u2019t want to spend more than \\$5. At the hamburger restaurant near his college, each hamburger has 240 calories and costs \\$1.40. Each cookie has 160 calories and costs \\$0.50.<\/p>\n<p id=\"fs-id1163872506375\"><span class=\"token\">a)<\/span>\u00a0Write a system of inequalities to model this situation.<span data-type=\"newline\">\n<\/span><span class=\"token\">b)<\/span>\u00a0Graph the system.<span data-type=\"newline\">\n<\/span><span class=\"token\">c)<\/span>\u00a0Could he eat 3 hamburgers and 1 cookie?<span data-type=\"newline\">\n<\/span><span class=\"token\">d)<\/span>\u00a0Could he eat 2 hamburgers and 4 cookies?<\/p>\n\n<\/div>\n<div id=\"fs-id1168345420640\" 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-id1163871975512\"><span class=\"token\">a)<\/span>\u00a0Let \\(h=\\) the number of hamburgers.<span data-type=\"newline\">\n<\/span>\\(\\phantom{\\rule{2.55em}{0ex}}c=\\) the number of cookies<span data-type=\"newline\">\n<\/span>To find the system of inequalities, translate the information.<span data-type=\"newline\">\n<\/span>The calories from hamburgers at 240 calories each, plus the calories from cookies at 160 calories each must be more that 800.<\/p>\n\n<div id=\"fs-id1168345415068\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(240h+160c\\ge 800\\)<\/div>\n<p id=\"fs-id1168034819709\">The amount spent on hamburgers at \\$1.40 each, plus the amount spent on cookies at \\$0.50 each must be no more than \\$5.00.<\/p>\n\n<div id=\"fs-id1168345398825\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\\(1.40h+0.50c\\le 5\\)<\/div>\n<p id=\"fs-id1168034819926\">We have our system of inequalities. \\(\\phantom{\\rule{3em}{0ex}}\\left\\{\\begin{array}{c}240h+160c\\ge 800\\hfill \\\\ 1.40h+0.50c\\le 5\\hfill \\end{array}\\)<\/p>\n<p id=\"fs-id1163872748969\"><span class=\"token\">b)<\/span><span data-type=\"newline\">\n<\/span><\/p>\n\n<table id=\"fs-id1167835489162\" class=\"unnumbered unstyled can-break\" summary=\"This figure has words and a graph. The words say, \u201cTo graph 240h + 160c is greater than or equal to 800, graph 240h + 160c = 800 as solid line. Choose (0, 0) as a test point. Since it does not make the inequality true, shade (red) the side that does not include the point (0, 0).\u201d The graph shows two solide lines. The area to the right of one line is shaded, and the area to the left of the other line is shaded. There is an area where to the two shaded areas overlap.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">To graph \\(240h+160c\\ge 800\\) graph \\(240h+160c=800\\) as a solid line.<span data-type=\"newline\">\n<\/span>Choose (0, 0) as a test point. it does not make the inequality true.<span data-type=\"newline\">\n<\/span>So, shade (red) the side that does not include the point (0, 0).<span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span><span data-type=\"newline\">\n<\/span>To graph \\(1.40h+0.50c\\le 5\\), graph \\(1.40h+0.50c=5\\) as a solid line.<span data-type=\"newline\">\n<\/span>Choose (0,0) as a test point. It makes the inequality true. So, shade<span data-type=\"newline\">\n<\/span>(blue) the side that includes the point.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167834376306\" data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_011a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span data-type=\"newline\">\n<\/span>The solution of the system is the region of the graph that is double shaded and so is shaded darker.\n<p id=\"fs-id1163872841697\"><span class=\"token\">c)<\/span>\u00a0To determine if 3 hamburgers and 2 cookies would meet Omar\u2019s criteria, we see if the point (3, 1) is in the solution region. It is. He might choose to eat 3 hamburgers and 2 cookies.<span data-type=\"newline\">\n<\/span><\/p>\n<span class=\"token\">d)<\/span>\u00a0To determine if 2 hamburgers and 4 cookies would meet Omar\u2019s criteria, we see if the point (2, 4) is in the solution region. It is. He might choose to eat 2 hamburgers and 4 cookies.\n<p id=\"fs-id1168345743092\">We could also test the possible solutions by substituting the values into each inequality.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">TRY IT 8<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345256115\" data-type=\"problem\">\n<p id=\"fs-id1168345376826\">Tension needs to eat at least an extra 1,000 calories a day to prepare for running a marathon. He has only \\$25 to spend on the extra food he needs and will spend it on \\$0.75 donuts which have 360 calories each and \\$2 energy drinks which have 110 calories.<\/p>\n<p id=\"fs-id1167835166563\">a) Write a system of inequalities that models this situation.\nb) Graph the system.\nc) Can he buy 8 donuts and 4 energy drinks?\nd) Can he buy 1 donut and 3 energy drinks?<\/p>\n\n<\/div>\n<div id=\"fs-id1168345229342\" data-type=\"solution\"><details open=\"open\"><summary class=\"answer\">Show answer<\/summary>a) \\(\\left\\{\\begin{array}{c}0.75d+2e\\le 25\\hfill \\\\ 360d+110e\\ge 1000\\hfill \\end{array}\\)\n\nb)\n<span id=\"fs-id1168345250261\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 0.75d + 2e is less than or equal to 25 and 360d + 110e is greater than or equal to 1000. The area to the left or right of each line is shaded slightly different colors with the overlapping area also shaded a slightly different color.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure05_06_025_img.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 0.75d + 2e is less than or equal to 25 and 360d + 110e is greater than or equal to 1000. The area to the left or right of each line is shaded slightly different colors with the overlapping area also shaded a slightly different color.\" data-media-type=\"image\/jpeg\"><\/span>\n\nc) yes\n\nd) no\n\n<\/details><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345367666\" class=\"media-2\" data-type=\"note\">\n<p id=\"fs-id1168345274887\">Access these online resources for additional instruction and practice with graphing systems of linear inequalities.<\/p>\n\n<ul id=\"fs-id1168345441091\" data-display=\"block\">\n \t<li><a href=\"http:\/\/www.openstax.org\/l\/25GSInequal1\">Graphical System of Inequalities<\/a><\/li>\n \t<li><a href=\"http:\/\/www.openstax.org\/l\/25GSInequal2\">Systems of Inequalities<\/a><\/li>\n \t<li><a href=\"http:\/\/www.openstax.org\/l\/25GSInequal3\">Solving Systems of Linear Inequalities by Graphing<\/a><\/li>\n<\/ul>\n<h1 data-type=\"title\">Key Concepts<\/h1>\n<ul id=\"fs-id1168345434983\" data-bullet-style=\"bullet\">\n \t<li><strong data-effect=\"bold\">To Solve a System of Linear Inequalities by Graphing<\/strong>\n<ol id=\"fs-id1169754231290\" class=\"stepwise\" type=\"1\">\n \t<li>Graph the first inequality.\n<ul id=\"fs-id1171792500552\" data-bullet-style=\"circled\">\n \t<li>Graph the boundary line.<\/li>\n \t<li>Shade in the side of the boundary line where the inequality is true.<\/li>\n<\/ul>\n<\/li>\n \t<li>On the same grid, graph the second inequality.\n<ul id=\"fs-id1171792759774\" data-bullet-style=\"circled\">\n \t<li>Graph the boundary line.<\/li>\n \t<li>Shade in the side of that boundary line where the inequality is true.<\/li>\n<\/ul>\n<\/li>\n \t<li>The solution is the region where the shading overlaps.<\/li>\n \t<li>Check by choosing a test point.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<div id=\"fs-id1168345271071\" class=\"practice-perfect\" data-depth=\"2\">\n<h1 data-type=\"title\">4.5 Exercise Set<\/h1>\n<p id=\"fs-id1169753987508\">In the following exercises, determine whether each ordered pair is a solution to the system.<\/p>\n\n<ol class=\"twocolumn\">\n \t<li><img class=\"alignnone wp-image-1407\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-1-300x122.png\" alt=\"\" width=\"138\" height=\"56\">\n<ol type=\"a\">\n \t<li>\\(\\left(3,-3\\right)\\)<\/li>\n \t<li>\\(\\left(7,1\\right)\\)<\/li>\n<\/ol>\n<\/li>\n \t<li><img class=\"alignnone wp-image-1408\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-2-300x128.png\" alt=\"\" width=\"141\" height=\"60\">\n<ol type=\"a\">\n \t<li>\\(\\left(6,-4\\right)\\)<\/li>\n \t<li>\\(\\left(3,0\\right)\\)<\/li>\n<\/ol>\n<\/li>\n \t<li><img class=\"alignnone wp-image-1409\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-3-300x115.png\" alt=\"\" width=\"141\" height=\"54\">\n<ol type=\"a\">\n \t<li>\\(\\left(2,3\\right)\\)<\/li>\n \t<li>\\(\\left(7,-1\\right)\\)<\/li>\n<\/ol>\n<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x+3y\\ge 2\\hfill \\\\ 4x-6y&lt;-1\\hfill \\end{array}\\)\n<ol type=\"a\">\n \t<li>\\(\\left(\\frac{3}{2},\\frac{4}{3}\\right)\\)<\/li>\n \t<li>\\(\\left(\\frac{1}{4},\\frac{7}{6}\\right)\\)<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p id=\"fs-id1169749956485\">In the following exercises, solve each system by graphing.<\/p>\n\n<ol class=\"twocolumn\" start=\"5\">\n \t<li><img class=\"alignnone wp-image-1410\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-5-300x144.png\" alt=\"\" width=\"140\" height=\"67\"><\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y&lt;2x-1\\hfill \\\\ y\\le -\\frac{1}{2}x+4\\hfill \\end{array}\\)<\/li>\n \t<li><img class=\"alignnone wp-image-1411\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-7-300x117.png\" alt=\"\" width=\"139\" height=\"54\"><\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}3x-y\\le 6\\hfill \\\\ y\\ge -\\frac{1}{2}x\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}2x-5y&lt;10\\hfill \\\\ 3x+4y\\ge 12\\hfill \\end{array}\\)<\/li>\n \t<li><img class=\"alignnone wp-image-1412\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-10-300x112.png\" alt=\"\" width=\"140\" height=\"52\"><\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x-2y&lt;3\\hfill \\\\ y\\le 1\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y\\ge -\\frac{1}{2}x-3\\hfill \\\\ x\\le 2\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y\\ge \\frac{3}{4}x-2\\hfill \\\\ y&lt;2\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}3x-4y&lt;8\\hfill \\\\ x&lt;1\\hfill \\end{array}\\)<\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}x\\ge 3\\hfill \\\\ y\\le 2\\hfill \\end{array}\\)<\/li>\n \t<li><img class=\"alignnone wp-image-1413 \" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-15.png\" alt=\"\" width=\"139\" height=\"54\"><\/li>\n \t<li><img class=\"alignnone wp-image-1414\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-17-300x109.png\" alt=\"\" width=\"140\" height=\"51\"><\/li>\n \t<li><img class=\"alignnone wp-image-1415\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-18-300x118.png\" alt=\"\" width=\"140\" height=\"55\"><\/li>\n \t<li>\\(\\left\\{\\begin{array}{c}y\\le -\\frac{1}{4}x-2\\hfill \\\\ x+4y&lt;6\\hfill \\end{array}\\)<\/li>\n \t<li><img class=\"alignnone wp-image-1416\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-20-300x110.png\" alt=\"\" width=\"141\" height=\"52\"><\/li>\n<\/ol>\n<p id=\"fs-id1169751892445\">In the following exercises, translate to a system of inequalities and solve.<\/p>\n\n<ol start=\"21\">\n \t<li>Caitlyn sells her drawings at the county fair. She wants to sell at least 60 drawings and has portraits and landscapes. She sells the portraits for \\$15 and the landscapes for \\$10. She needs to sell at least \\$800 worth of drawings in order to earn a profit.\n<ol type=\"a\">\n \t<li>Write a system of inequalities to model this situation.<\/li>\n \t<li>Graph the system.<\/li>\n \t<li>Will she make a profit if she sells 20 portraits and 35 landscapes?<\/li>\n \t<li>Will she make a profit if she sells 50 portraits and 20 landscapes?<\/li>\n<\/ol>\n<\/li>\n \t<li>Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than \\$500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is \\$3 and for a package the cost is \\$7.\n<ol type=\"a\">\n \t<li>Write a system of inequalities to model this situation.<\/li>\n \t<li>Graph the system.<\/li>\n \t<li>Can she mail 60 cards and 26 packages?<\/li>\n \t<li>Can she mail 90 cards and 40 packages?<\/li>\n<\/ol>\n<\/li>\n \t<li>Jocelyn is pregnant and needs to eat at least 500 more calories a day than usual. When buying groceries one day with a budget of \\$15 for the extra food, she buys bananas that have 90 calories each and chocolate granola bars that have 150 calories each. The bananas cost \\$0.35 each and the granola bars cost \\$2.50 each.\n<ol type=\"a\">\n \t<li>Write a system of inequalities to model this situation.<\/li>\n \t<li>Graph the system.<\/li>\n \t<li>Could she buy 5 bananas and 6 granola bars?<\/li>\n \t<li>Could she buy 3 bananas and 4 granola bars?<\/li>\n<\/ol>\n<\/li>\n \t<li>Jocelyn desires to increase both her protein consumption and caloric intake. She desires to have at least 35 more grams of protein each day and no more than an additional 200 calories daily. An ounce of cheddar cheese has 7 grams of protein and 110 calories. An ounce of parmesan cheese has 11 grams of protein and 22 calories.\n<ol type=\"a\">\n \t<li>Write a system of inequalities to model this situation.<\/li>\n \t<li>Graph the system.<\/li>\n \t<li>Could she eat 1 ounce of cheddar cheese and 3 ounces of parmesan cheese?<\/li>\n \t<li>Could she eat 2 ounces of cheddar cheese and 1 ounce of parmesan cheese?<\/li>\n<\/ol>\n<\/li>\n \t<li>Tickets for an American Baseball League game for 3 adults and 3 children cost less than ?75, while tickets for 2 adults and 4 children cost less than \\$62.\n<ol type=\"a\">\n \t<li>Write a system of inequalities to model this problem.<\/li>\n \t<li>Graph the system.<\/li>\n \t<li>Could the tickets cost \\$20 for adults and \\$8 for children?<\/li>\n \t<li>Could the tickets cost \\$15 for adults and \\$5 for children?<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\">\n<h1 id=\"fs-id1169752082080\">Answers:<\/h1>\n<ol class=\"twocolumn\">\n \t<li style=\"list-style-type: none;\">\n<ol type=\"a\">\n \t<li>true<\/li>\n \t<li>false<\/li>\n<\/ol>\n<ol type=\"a\">\n \t<li>false<\/li>\n \t<li>true<\/li>\n<\/ol>\n<ol type=\"a\">\n \t<li>true<\/li>\n \t<li>false<\/li>\n<\/ol>\n<ol type=\"a\">\n \t<li>true<\/li>\n \t<li>true<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345276530\" data-type=\"problem\">\n\n5.\n\n<\/div>\n<div id=\"fs-id1168345273986\" data-type=\"solution\"><span id=\"fs-id1168345708305\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is less than or equal to 3x + 2 and y is greater than x \u2013 1. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_201_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is less than or equal to 3x + 2 and y is greater than x \u2013 1. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\" data-media-type=\"image\/jpeg\"><\/span><\/div><\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168341851439\" data-type=\"problem\">\n\n6.\n\n<\/div>\n<div id=\"fs-id1168345634904\" data-type=\"solution\"><span id=\"fs-id1168345623920\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is less than 2x - 1 and y is less than or equal to -(1\/2)x + 4. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_203_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is less than 2x - 1 and y is less than or equal to -(1\/2)x + 4. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\" data-media-type=\"image\/jpeg\"><\/span><\/div><\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345462056\" data-type=\"exercise\">\n<div id=\"fs-id1168345301976\" data-type=\"problem\">\n<p id=\"fs-id1168345535241\">7.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1168345418364\" data-type=\"exercise\">\n<div id=\"fs-id1168341917954\" data-type=\"solution\"><span id=\"fs-id1168345434088\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of x \u2013 y is greater than 1 and y is less than \u2013(1\/4)x + 3. The area to the right or below each line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_205_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of x \u2013 y is greater than 1 and y is less than \u2013(1\/4)x + 3. The area to the right or below each line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\" data-media-type=\"image\/jpeg\"><\/span><\/div>\n<\/div><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345274250\" data-type=\"problem\">\n\n8.\n\n<\/div>\n<div id=\"fs-id1168345346375\" data-type=\"solution\"><span id=\"fs-id1168345647738\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 3x \u2013 y is less than or equal to 6 and y is greater than or equal to \u2013(1\/2)x. The area to the right or above each line is shaded different colors with the overlapping area also shaded a different color.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_207_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 3x \u2013 y is less than or equal to 6 and y is greater than or equal to \u2013(1\/2)x. The area to the right or above each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\"><\/span><\/div><\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345432909\" data-type=\"problem\">\n\n9.\n\n<\/div>\n<div id=\"fs-id1168345540531\" data-type=\"solution\"><span id=\"fs-id1168345560367\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 2x \u2013 5y is less than 10 and 3x +4y is greater than or equal to 12. The area to the right above each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_209_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 2x \u2013 5y is less than 10 and 3x +4y is greater than or equal to 12. The area to the right above each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\" data-media-type=\"image\/jpeg\"><\/span><\/div><\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345254775\" data-type=\"problem\">\n\n10.\n\n<\/div>\n<div id=\"fs-id1168341960737\" data-type=\"solution\"><span id=\"fs-id1168341852659\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 2x + 2y is greater than -4 and \u2013x + 3y is greater than or equal to 9. The area to the right or above each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_211_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 2x + 2y is greater than -4 and \u2013x + 3y is greater than or equal to 9. The area to the right or above each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\" data-media-type=\"image\/jpeg\"><\/span><\/div><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345217910\" data-type=\"problem\">\n\n11.\n\n<\/div>\n<div id=\"fs-id1168345744963\" data-type=\"solution\"><span id=\"fs-id1168345511763\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of x \u2013 2y is less than 3 and y is less than or equal to 1. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_213_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of x \u2013 2y is less than 3 and y is less than or equal to 1. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\" data-media-type=\"image\/jpeg\"><\/span><\/div><\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345633331\" data-type=\"problem\">\n\n12.\n\n<\/div>\n<div id=\"fs-id1168345644626\" data-type=\"solution\"><span id=\"fs-id1168345292038\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to (-1\/2)x - 3 and x is less than or equal to 2. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_215_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to (-1\/2)x - 3 and x is less than or equal to 2. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\"><\/span><\/div><\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345540621\" data-type=\"problem\">\n\n13.\n\n<\/div>\n<div id=\"fs-id1168345387414\" data-type=\"solution\"><span id=\"fs-id1168341917038\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to (3\/4)x - 2 and y is less than 2. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_217_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to (3\/4)x - 2 and y is less than 2. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\" data-media-type=\"image\/jpeg\"><\/span><\/div><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345387655\" data-type=\"problem\">\n\n14.\n\n<\/div>\n<div id=\"fs-id1168341864160\" data-type=\"solution\"><span data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 3x \u2013 4y is less than 8 and x is less than 1. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_219_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 3x \u2013 4y is less than 8 and x is less than 1. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\" data-media-type=\"image\/jpeg\"><\/span><\/div><\/td>\n<td style=\"width: 33.3333%;\">\n<div data-type=\"problem\">\n\n15.\n\n<\/div>\n<div id=\"fs-id1168345738042\" data-type=\"solution\"><span id=\"fs-id1168345214653\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of x is greater than or equal to 3 and y less than or equal to 2. The area to the right or below each line is shaded different colors with the overlapping area also shaded a different color.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_221_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of x is greater than or equal to 3 and y less than or equal to 2. The area to the right or below each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\"><\/span><\/div><\/td>\n<td style=\"width: 33.3333%;\">\n<p id=\"fs-id1168345439795\">16. No solution<span data-type=\"newline\">\n<\/span><\/p>\n<span id=\"fs-id1168345644259\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 2x + 4y is greater than 4 and y is less than or equal to (-1\/2)x - 2. The area to the left or right of each line is shaded different colors. There is no area where the shaded areas overlap. One line is dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_223_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 2x + 4y is greater than 4 and y is less than or equal to (-1\/2)x - 2. The area to the left or right of each line is shaded different colors. There is no area where the shaded areas overlap. One line is dotted.\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">17. No solution<span data-type=\"newline\">\n<\/span><span id=\"fs-id1168345434496\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of -2x + 6y is less than 0 and 6y is greater than 2x + 4. The area to the left or right of each line is shaded different colors. There is no area where the shaded areas overlap. Both lines are dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_225_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of -2x + 6y is less than 0 and 6y is greater than 2x + 4. The area to the left or right of each line is shaded different colors. There is no area where the shaded areas overlap. Both lines are dotted.\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168341958605\" data-type=\"problem\">\n\n18.\n\n<\/div>\n<div id=\"fs-id1168345429012\" data-type=\"solution\"><span id=\"fs-id1168345398337\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to -3x + 2 and 3x + y is greater than 5. The area to the right of each line is shaded different colors. One line is within the shaded area of the other. One line is dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_227_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to -3x + 2 and 3x + y is greater than 5. The area to the right of each line is shaded different colors. One line is within the shaded area of the other. One line is dotted.\" data-media-type=\"image\/jpeg\"><\/span><\/div><\/td>\n<td style=\"width: 33.3333%;\">\n<p id=\"fs-id1168345560336\">19.\\(x+4y&lt;6\\)<span data-type=\"newline\">\n<\/span><\/p>\n<span data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is less than or equal to (negative 1\/4)x \u2013 2 and x + 4y is less than 6. The area below each line is shaded different colors. One line is within the shaded area of the other. One line is dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_229_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is less than or equal to (negative 1\/4)x \u2013 2 and x + 4y is less than 6. The area below each line is shaded different colors. One line is within the shaded area of the other. One line is dotted.\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<p id=\"fs-id1168345638227\">20. \\(-2x+6y\\) &gt; \\(8\\)<span data-type=\"newline\">\n<\/span><\/p>\n<span id=\"fs-id1168345560952\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 3y is greater than x + 2 and -2x + 6y is greater than 8. The area above each line is shaded different colors. One line is within the shaded area of the other. Both lines are dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_231_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 3y is greater than x + 2 and -2x + 6y is greater than 8. The area above each line is shaded different colors. One line is within the shaded area of the other. Both lines are dotted.\" data-media-type=\"image\/jpeg\"><\/span><\/td>\n<td style=\"width: 33.3333%;\"><span class=\"token\">21. a) <\/span>\\(\\left\\{\\begin{array}{c}p+l\\ge 60\\hfill \\\\ 15p+10l\\ge 800\\hfill \\end{array}\\)\n\n<span class=\"token\">b)<\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1168345692981\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of p + l is greater than or equal to 60 and 15p + 10l is greater than or equal to 800. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_233_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of p + l is greater than or equal to 60 and 15p + 10l is greater than or equal to 800. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<span class=\"token\">c)<\/span> No\n\n<span class=\"token\">d)<\/span> Yes<\/td>\n<td style=\"width: 33.3333%;\"><span class=\"token\">22. a) <\/span>\\(\\left\\{\\begin{array}{c}7p+3c\\le 500\\hfill \\\\ p\\ge 2c+4\\hfill \\end{array}\\)\n\n<span class=\"token\">b)<\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1168345529466\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 7p + 3c is less than or equal to 500 and p is greater than or equal to 2c + 4. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_235_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 7p + 3c is less than or equal to 500 and p is greater than or equal to 2c + 4. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<span class=\"token\">c)<\/span> Yes\n\n<span class=\"token\">d)<\/span> No<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\"><span class=\"token\">23. a) <\/span>\\(\\left\\{\\begin{array}{c}90b+150g\\ge 500\\hfill \\\\ 0.35b+2.50g\\le 15\\hfill \\end{array}\\)\n\n<span data-type=\"newline\">b)\n<\/span><span id=\"fs-id1168345287674\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 90b + 150g is greater than or equal to 500 and 0.35b + 2.50g is less than or equal to 15. The area to the right or below each line is shaded different colors with the overlapping area also shaded a different color.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_237_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 90b + 150g is greater than or equal to 500 and 0.35b + 2.50g is less than or equal to 15. The area to the right or below each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<span class=\"token\">c)<\/span> No\n\n<span class=\"token\">d)<\/span> Yes<\/td>\n<td style=\"width: 33.3333%;\"><span class=\"token\">24. a) <\/span>\\(\\left\\{\\begin{array}{c}7c+11p\\ge 35\\hfill \\\\ 110\\text{c}+22p\\le 200\\hfill \\end{array}\\)\n\n<span class=\"token\">b)<\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1168345448185\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 7c + 11p is greater than or equal to 35 and 110c + 22p is less than or equal to 200. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_239_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 7c + 11p is greater than or equal to 35 and 110c + 22p is less than or equal to 200. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<span class=\"token\">c)<\/span> Yes\n\n<span class=\"token\">d)<\/span> No<\/td>\n<td style=\"width: 33.3333%;\"><span class=\"token\">25. a) <\/span>\\(\\left\\{\\begin{array}{c}3a+3c&lt;75\\hfill \\\\ 2\\text{a}+4c&lt;62\\hfill \\end{array}\\)\n\n<span class=\"token\">b)<\/span><span data-type=\"newline\">\n<\/span><span id=\"fs-id1168341919046\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 3a + 3c is less than 75 and 2a + 4c is less than 62. The area to the left ofeach line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\"><img src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_241_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 3a + 3c is less than 75 and 2a + 4c is less than 62. The area to the left ofeach line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\" data-media-type=\"image\/jpeg\"><\/span>\n\n<span class=\"token\">c)<\/span> No\n\n<span class=\"token\">d)<\/span> Yes<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\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>Determine whether an ordered pair is a solution of a system of linear inequalities<\/li>\n<li>Solve a system of linear inequalities by graphing<\/li>\n<li>Solve applications of systems of inequalities<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345636018\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities<\/h1>\n<p id=\"fs-id1168344338607\">The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.<\/p>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">System of Linear Inequalities<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Two or more linear inequalities grouped together form a <span data-type=\"term\">system of linear inequalities<\/span>.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1168345550211\">A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown below.<\/p>\n<div id=\"fs-id1168344523603\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c7ccc7d8506205cc0bbc13b242bfb269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#52;&#121;&#92;&#103;&#101;&#32;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#60;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"fs-id1168341972977\">To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> that make both inequalities true.<\/p>\n<div 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\">Solutions of a System of Linear Inequalities<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true.<\/p>\n<\/div>\n<\/div>\n<p><span style=\"text-align: initial; font-size: 14pt;\">The solution of a system of linear inequalities is shown as a shaded region in the <\/span><em style=\"text-align: initial; font-size: 14pt;\" data-effect=\"italics\">x-y<\/em><span style=\"text-align: initial; font-size: 14pt;\"> coordinate system that includes all the points whose ordered pairs make the inequalities true.<\/span><\/p>\n<\/div>\n<\/div>\n<p>To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.<\/p>\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 data-type=\"problem\">\n<p id=\"fs-id1168345198719\">Determine whether the ordered pair is a solution to the system. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c7ccc7d8506205cc0bbc13b242bfb269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#52;&#121;&#92;&#103;&#101;&#32;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#60;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1168345397911\"><span class=\"token\">a)<\/span> (\u22122, 4) b) (3,1)<\/p>\n<\/div>\n<div id=\"fs-id1168345440026\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<p>&nbsp;<\/p>\n<p>a) Is the ordered pair (\u22122, 4) a solution?<span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1168345428380\" data-type=\"media\" data-alt=\"This figure says, \u201cWe substitute x = -2 and y = 4 into both inequalities. The first inequality, x + 4 y is greater than or equal to 10 becomes -2 plus 4 times 4 is greater than or less than 10 or 14 is great than or less than 10 which is true. The second inequality, 3x \u2013 2y is less than 12 becomes 3 times -2 \u2013 2 times 4 is less than 12 or -14 is less than 12 which is true.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2019\/07\/CNX_ElemAlg_Figure_05_06_001_img_new.jpg\" alt=\"This figure says, \u201cWe substitute x = -2 and y = 4 into both inequalities. The first inequality, x + 4 y is greater than or equal to 10 becomes -2 plus 4 times 4 is greater than or less than 10 or 14 is great than or less than 10 which is true. The second inequality, 3x \u2013 2y is less than 12 becomes 3 times -2 \u2013 2 times 4 is less than 12 or -14 is less than 12 which is true.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1167834065902\">The ordered pair (\u22122, 4) made both inequalities true. Therefore (\u22122, 4) is a solution to this system.<\/p>\n<p>b) Is the ordered pair (3,1) a solution?<span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1168345417372\" data-type=\"media\" data-alt=\"This figure says, \u201cWe substitute x 3 and y = 1 into both inequalities.\u201d The first inequality, x + 4y is greater than or equal to 10 becomes 3 + 4 times 1 is greater than or equal to 10 or y is greater than or equal to 10 which is false. The second inequality, 3x -2y is less than 12 becomes 3 times 3 \u2013 two times 1 is less than 12 or 7 is less than 12 which is true.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_002_img_new.jpg\" alt=\"This figure says, \u201cWe substitute x 3 and y = 1 into both inequalities.\u201d The first inequality, x + 4y is greater than or equal to 10 becomes 3 + 4 times 1 is greater than or equal to 10 or y is greater than or equal to 10 which is false. The second inequality, 3x -2y is less than 12 becomes 3 times 3 \u2013 two times 1 is less than 12 or 7 is less than 12 which is true.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p id=\"fs-id1168345327228\">The ordered pair (3,1) made one inequality true, but the other one false. Therefore (3,1) is not a solution to this system.<\/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 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Determine whether the ordered pair is a solution to the system.<\/p>\n<div id=\"fs-id1168345253745\" data-type=\"note\">\n<div id=\"fs-id1168345510038\" data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1168345191853\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1381\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.07.43-PM-300x120.png\" alt=\"\" width=\"140\" height=\"56\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.07.43-PM-300x120.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.07.43-PM-1024x409.png 1024w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.07.43-PM-768x307.png 768w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.07.43-PM-65x26.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.07.43-PM-225x90.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.07.43-PM-350x140.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.07.43-PM.png 1082w\" sizes=\"auto, (max-width: 140px) 100vw, 140px\" \/><\/p>\n<p id=\"fs-id1168345436310\">a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0d66a71b8940b998e4f29f8cccda06d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5a00a53e3c5401663e307951ae0d74c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345556267\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345745212\">a) no b) yes<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve a System of Linear Inequalities by Graphing<\/h1>\n<p id=\"fs-id1168345415085\">The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.<\/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 data-type=\"title\">How to Solve a System of Linear inequalities<\/div>\n<div id=\"fs-id1168345385725\" data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1168345384150\">Solve the system by graphing.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-7c9c6f0f00c1f3c1dcc7d3d2e913bff7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#50;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"101\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div 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><span id=\"fs-id1168345191155\" data-type=\"media\" data-alt=\"This is a table with three columns and several rows. The first row says, \u201cStep 1: Graph the first inequality. We will graph y is greater than or equal to 2x \u2013 1.\u201d There are two equations givens, y is greater than or equal to 2x \u2013 1 and y is less than x + 1. The table then reads, \u201cGraph the boundary line. We graph the line y = 2x \u2013 1. It is a solid line because the inequality sign is greater than or equal to. Shade in the side of the boundary line where the inequality is true. We choose (0, 0) as a test point. It is a solution to y is greater than or equal to 2x \u2013 1, so we shad in the left side of the boundary line.\u201d There is a figure of a line graphed on an x y coordinate plane. The area to the left of the line is shaded.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_003a_img_new.jpg\" alt=\"This is a table with three columns and several rows. The first row says, \u201cStep 1: Graph the first inequality. We will graph y is greater than or equal to 2x \u2013 1.\u201d There are two equations givens, y is greater than or equal to 2x \u2013 1 and y is less than x + 1. The table then reads, \u201cGraph the boundary line. We graph the line y = 2x \u2013 1. It is a solid line because the inequality sign is greater than or equal to. Shade in the side of the boundary line where the inequality is true. We choose (0, 0) as a test point. It is a solution to y is greater than or equal to 2x \u2013 1, so we shad in the left side of the boundary line.\u201d There is a figure of a line graphed on an x y coordinate plane. The area to the left of the line is shaded.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345409450\" data-type=\"media\" data-alt=\"The second row then says, \u201cStep 2: On the same grid, graph the second inequality. We will graph y is less than x + 1 on the same grid. Grph the boundary line. We graph the lin y = x + 1. It is a dashed line because the inequality sign is less than. There is a graph which shows two lines graphed on an x y coordinate plane. The area to the left of one line is shaded. The area to the right of the second line is shaded. There is a small area where the shaded areas overlap. The table then says, \u201cShade in the side of that boundary line where the inequality is true. Again we use (0, 0) as a test point. It is a solution so we shade in that side of the line y = x + 1.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_003b_img_new.jpg\" alt=\"The second row then says, \u201cStep 2: On the same grid, graph the second inequality. We will graph y is less than x + 1 on the same grid. Grph the boundary line. We graph the lin y = x + 1. It is a dashed line because the inequality sign is less than. There is a graph which shows two lines graphed on an x y coordinate plane. The area to the left of one line is shaded. The area to the right of the second line is shaded. There is a small area where the shaded areas overlap. The table then says, \u201cShade in the side of that boundary line where the inequality is true. Again we use (0, 0) as a test point. It is a solution so we shade in that side of the line y = x + 1.\" data-media-type=\"image\/jpeg\" \/><\/span><span id=\"fs-id1168345209236\" data-type=\"media\" data-alt=\"The third row then says, \u201cStep 3: The solution is the region where the shading overlaps. The poing where the boundary lines intersect is not a solution because it is not a solution to y is less than x + 1. The solution is all points in the purple shaded region.\u201d\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_003c_img_new.jpg\" alt=\"The third row then says, \u201cStep 3: The solution is the region where the shading overlaps. The poing where the boundary lines intersect is not a solution because it is not a solution to y is less than x + 1. The solution is all points in the purple shaded region.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><span data-type=\"media\" data-alt=\"The fourth row then says, \u201cStep 4: Check by choosing a test point. We\u2019ll use (-1, -1) as a test point. Is (-1, -1) a solution to y is greater than or equal to 2x \u2013 1? -1 is greater than or equal to 2 times -1 \u2013 1 or -1 is greater than or equal to -3 true.\u201d\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_003d_img_new.jpg\" alt=\"The fourth row then says, \u201cStep 4: Check by choosing a test point. We\u2019ll use (-1, -1) as a test point. Is (-1, -1) a solution to y is greater than or equal to 2x \u2013 1? -1 is greater than or equal to 2 times -1 \u2013 1 or -1 is greater than or equal to -3 true.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\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 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p>Solve the system by graphing. <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1386\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-try-it-2-300x134.png\" alt=\"\" width=\"141\" height=\"63\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-try-it-2-300x134.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-try-it-2-65x29.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-try-it-2-225x100.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-try-it-2-350x156.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-try-it-2.png 768w\" sizes=\"auto, (max-width: 141px) 100vw, 141px\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345374077\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p><span id=\"fs-id1168345455090\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is less than 3x +2 and y is greater than \u2013x \u2013 1. The area to the right of each line is shaded slightly different colors with the overlapping area also shaded a slightly different color. Both lines are dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_013_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is less than 3x +2 and y is greater than \u2013x \u2013 1. The area to the right of each line is shaded slightly different colors with the overlapping area also shaded a slightly different color. Both lines are dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\"><span style=\"font-size: 14pt;\">Solve a system of linear inequalities by graphing.<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol id=\"fs-id1169753926787\" class=\"stepwise\" type=\"1\">\n<li>Graph the first inequality.\n<ul id=\"fs-id1168345277927\" data-bullet-style=\"bullet\">\n<li>Graph the boundary line.<\/li>\n<li>Shade in the side of the boundary line where the inequality is true.<\/li>\n<\/ul>\n<\/li>\n<li>On the same grid, graph the second inequality.\n<ul id=\"fs-id1168345399311\" data-bullet-style=\"bullet\">\n<li>Graph the boundary line.<\/li>\n<li>Shade in the side of that boundary line where the inequality is true.<\/li>\n<\/ul>\n<\/li>\n<li>The solution is the region where the shading overlaps.<\/li>\n<li>Check by choosing a test point.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"howto\" data-type=\"note\">\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 id=\"fs-id1168341961505\" data-type=\"problem\">\n<p>Solve the system by graphing. <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1388\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.11.42-PM-300x154.png\" alt=\"\" width=\"140\" height=\"72\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.11.42-PM-300x154.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.11.42-PM-768x395.png 768w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.11.42-PM-65x33.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.11.42-PM-225x116.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.11.42-PM-350x180.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.11.42-PM.png 918w\" sizes=\"auto, (max-width: 140px) 100vw, 140px\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167835369328\" style=\"width: 100%;\" summary=\"This figure has both words and a graph. The words say, \u201cGraph x \u2013 y is greater than 3, by graphing x \u2013 y = 3 and testing a point. The intercepts are x = 3 and y = -3 and the boundary line will be dashed. Test (0, 0) which makes the inequality false so shade (red) the side that does not contain (0, 0).\u201d The graph shows a line graphed on an x y-coordinate plane. The line is dashed and the area to the right of the line is shaded.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 58.4091%;\" data-valign=\"top\">Graph\u00a0<em data-effect=\"italics\">x<\/em>\u00a0\u2212\u00a0<em data-effect=\"italics\">y<\/em>\u00a0&gt; 3, by graphing\u00a0<em data-effect=\"italics\">x<\/em>\u00a0\u2212\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= 3 and<span data-type=\"newline\"><br \/>\n<\/span>testing a point.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>The intercepts are\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 3 and\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= \u22123 and the boundary<span data-type=\"newline\"><br \/>\n<\/span>line will be dashed.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Test (0, 0). It makes the inequality false. So,<span data-type=\"newline\"><br \/>\n<\/span>shade the side that does not contain (0, 0) red.<\/td>\n<td style=\"width: 41.4773%;\" data-valign=\"top\"><span id=\"fs-id1167831213870\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_004a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/p>\n<p><span id=\"fs-id1167831213870\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_004a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 58.4091%;\" data-valign=\"top\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-977233750791ab2a40a83a40211f8c02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -6px;\" \/> by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-167ff240d536e8f45621849d73479973_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#120;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"98\" style=\"vertical-align: -6px;\" \/><span data-type=\"newline\"><br \/>\n<\/span>using the slope <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a0e35a4e0a70aabb2f4ed03eca173840_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/> and\u00a0<em data-effect=\"italics\">y<\/em>\u2212intercept<span data-type=\"newline\"><br \/>\n<\/span><em data-effect=\"italics\">b<\/em>\u00a0= 4. The boundary line will be dashed.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Test (0, 0). It makes the inequality true, so shade the side that contains (0, 0) blue.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Choose a test point in the solution and verify that it is a solution to both inequalities.<\/td>\n<td style=\"width: 41.4773%;\"><span id=\"fs-id1167831213870\" data-type=\"media\" data-alt=\".\"><span id=\"fs-id1167835369155\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_005a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The point of intersection of the two lines is not included as both boundary lines were dashed. The solution is the area shaded twice which is the darker-shaded region.<\/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 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"note\">\n<p>Solve the system by graphing. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e0e28aaed5e44520fa9e00bce2f620d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#92;&#108;&#101;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"103\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168341906364\" data-type=\"note\">\n<div data-type=\"exercise\">\n<div id=\"fs-id1168345260430\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p><span id=\"fs-id1168345436372\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of x + y is less than or equal to 2 and y is greater than or equal to (2\/3)x \u2013 1. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_027_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of x + y is less than or equal to 2 and y is greater than or equal to (2\/3)x \u2013 1. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"solution\">\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-id1168345337187\" data-type=\"problem\">\n<p id=\"fs-id1168345385902\">Solve the system by graphing. <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1392\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.14.43-PM-300x147.png\" alt=\"\" width=\"141\" height=\"69\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.14.43-PM-300x147.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.14.43-PM-768x377.png 768w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.14.43-PM-65x32.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.14.43-PM-225x110.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.14.43-PM-350x172.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.14.43-PM.png 774w\" sizes=\"auto, (max-width: 141px) 100vw, 141px\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168341864081\" data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167835357015\" class=\"unnumbered unstyled can-break\" summary=\"This figure has both words and a graph. The words say, \u201cGraph x \u2013 2y is less than 5, by graphing x \u2013 2y = 5 and testing a point. The intercepts are x = 5 and y = -2.5 and the boundary line will be dashed. Test (0, 0) which makes the inequality true, so shade (red) the side that contains (0, 0).\u201d The figure shows a dashed line graphed on an x y-coordinate plane. The area above the line is shaded.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b909669d87cdc887dd5b7e2cdc79d534_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#60;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\" \/>, by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0f592894dfc669be1583facabdf7c38b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\" \/> and testing a point.<span data-type=\"newline\"><br \/>\n<\/span>The intercepts are\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 5 and\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= \u22122.5 and the boundary line will be dashed.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Test (0, 0). It makes the inequality true. So, shade the side<span data-type=\"newline\"><br \/>\n<\/span>that contains (0, 0) red.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167831911196\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_006a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>Graph\u00a0<em data-effect=\"italics\">y<\/em>\u00a0&gt; \u22124, by graphing\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= \u22124 and recognizing that it is a<span data-type=\"newline\"><br \/>\n<\/span>horizontal line through\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= \u22124. The boundary line will be dashed.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Test (0, 0). It makes the inequality true. So, shade (blue)<span data-type=\"newline\"><br \/>\n<\/span>the side that contains (0, 0) blue.<\/td>\n<td><span id=\"fs-id1167831911196\" data-type=\"media\" data-alt=\".\"><span id=\"fs-id1167831896589\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_007a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345255474\">The point (0, 0) is in the solution and we have already found it to be a solution of each inequality. The point of intersection of the two lines is not included as both boundary lines were dashed.<\/p>\n<p>The solution is the area shaded twice which is the darker-shaded region.<\/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 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div data-type=\"problem\">\n<p id=\"fs-id1168345325877\">Solve the system by graphing. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3966a961ce10d55c8bf0a7cfc1d2450a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#51;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"101\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345390829\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_028_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to 3x - 2 and y is less than -1. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\" data-media-type=\"image\/jpeg\" \/><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"text-align: initial; font-size: 14pt;\">Systems of linear inequalities where the boundary lines are parallel might have no solution. We\u2019ll see this in the next example<\/span><span style=\"text-align: initial; font-size: 14pt;\">.<\/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-id1168345241928\" data-type=\"problem\">\n<p id=\"fs-id1168345743048\">Solve the system by graphing. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ff42697ceaa606e69a472609a1507cf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#43;&#51;&#121;&#92;&#103;&#101;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167834279222\" class=\"unnumbered unstyled can-break\" summary=\"This figure has both words and a graph. The first set of words says, \u201cGraph 4x + 3y is greater than or equal to 12, by graphing 4x + 3 = 12 and testing a point. The intercepts are x = 3 and y = 4 and the boundary line will be solid. Test (0, 0) which makes the inequality false, s oshade (blue) the side that does not contain (0, 0).\u201d The figure shows a line graphed on an x y-coordinate plane. The line is solid, and the area to the right of the line is shaded. The second set of words says, \u201cGraph y is less than (-4\/3)x + 1 by graphing y = (-4\/3)x + 1 using the slope m = (-4\/3) and the y-intercept b = 1. The boundarly line will be dashed. Test (0, 0) which makes the inequality true, so shade (red) the side that contains (0, 0). The figure shows two line graphed on an x y-coordinate plane. One line is solid, and the area to the right is shaded. The other line is dashed, and the area to the left is shaded. There is no area of overlap between the two shaded areas.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3e70c1e4d420dcf42029a4a07afb46bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#51;&#121;&#92;&#103;&#101;&#32;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/>, by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-78fad9dbf93f5bbe1e94c2ac7a1f17ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#51;&#121;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/> and testing a point.<span data-type=\"newline\"><br \/>\n<\/span>The intercepts are\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= 3 and\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= 4 and the boundary line will be solid.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Test (0, 0). It makes the inequality false. So,<span data-type=\"newline\"><br \/>\n<\/span>shade the side that does not contain (0, 0) red.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167834423065\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_008a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-99a9f07887b88c4b3874909153d098e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#60;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/> by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-08a48b76f35b590c7fe96f2bd1fee4d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -6px;\" \/> using the<span data-type=\"newline\"><br \/>\n<\/span>slope <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-96dd2b776d0c9ab0073c0d1943e82b3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"48\" style=\"vertical-align: -6px;\" \/> and the\u00a0<em data-effect=\"italics\">y<\/em>-intercept\u00a0<em data-effect=\"italics\">b<\/em>\u00a0= 1. The boundary line will be dashed.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Test (0, 0). It makes the inequality true. So,<span data-type=\"newline\"><br \/>\n<\/span>shade the side that contains (0, 0) blue.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167831893018\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_008b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345360332\">There is no point in both shaded regions, so the system has no solution. This system has no solution.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\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-id1168345190544\" data-type=\"problem\">\n<p id=\"fs-id1168345407920\">Solve the system by graphing. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9c348ef299e4050b4df0ee5545573c0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#50;&#121;&#92;&#108;&#101;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p id=\"fs-id1168345260108\">no solution<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_029_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 3x \u2013 2y is less than or equal 12 and y is greater than or equal to (3\/2)x + 1. The area to the left or right of each line is shaded different colors. There is not overlapping area.\" data-media-type=\"image\/jpeg\" \/><\/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 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345230029\" data-type=\"problem\">\n<p>Solve the system by graphing. <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1399\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.16.28-PM-300x125.png\" alt=\"\" width=\"142\" height=\"59\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.16.28-PM-300x125.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.16.28-PM-768x320.png 768w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.16.28-PM-65x27.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.16.28-PM-225x94.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.16.28-PM-350x146.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/Screenshot-2021-08-11-at-12.16.28-PM.png 864w\" sizes=\"auto, (max-width: 142px) 100vw, 142px\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<table id=\"fs-id1167835330830\" style=\"width: 100%;\" summary=\"This figure has both words and a graph. The first set of words says, \u201cGraph y is greater than (1\/2)x \u2013 4 by graphing y = (1\/2)x \u2013 4 using the slope m = 1\/2 and the intercept b = -4. The boundary line will be dashed. Tess (0, 0) which makes the inequality true, so shade (red) the side that contains (0, 0).\u201d The graph shows a dashed line graphed on an x y-coordinate plane with the area above it shaded. The second set of words says, \u201cGraph x -2 y is less than -4, by graphing x \u2013 2y = -4 and testing a point. The intercepts are x = -4 and y = 2 and the boundary line will be dashed. Choose a test point in the solution and verify that it is a solution to both inequalities.\u201d The graph shows to lines graphed. One is dashed, and the area above it is shaded. The other line is solid, and the area above it is shaded. There is an area where the shaded areas overlap.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> &gt; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-a114fc6964433392a9a771022f1f56bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/> by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-52fb6a706fbcd64f2a7398346ce16a3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"85\" style=\"vertical-align: -6px;\" \/><span data-type=\"newline\"><br \/>\n<\/span>using the slope <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f4e7c1468d71a3131e728a60e2da9237_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"48\" style=\"vertical-align: -6px;\" \/> and the intercept<span data-type=\"newline\"><br \/>\n<\/span><em data-effect=\"italics\">b<\/em>\u00a0= \u22124. The boundary line will be dashed.<span data-type=\"newline\"><br \/>\n<\/span>Test (0, 0). It makes the inequality true. So,<span data-type=\"newline\"><br \/>\n<\/span>shade the side that contains (0, 0) red.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167834110177\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_009a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-864074ac6c923db53d28749a677e43cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#60;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" style=\"vertical-align: -4px;\" \/> by graphing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-46b8bb6abd0b3a1f2b13e61ff685657e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#61;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" style=\"vertical-align: -4px;\" \/> and testing a point.<span data-type=\"newline\"><br \/>\n<\/span>The intercepts are\u00a0<em data-effect=\"italics\">x<\/em>\u00a0= \u22124 and\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= 2 and the boundary<span data-type=\"newline\"><br \/>\n<\/span>line will be dashed.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>Choose a test point in the solution and verify<span data-type=\"newline\"><br \/>\n<\/span>that it is a solution to both inequalities.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167835309321\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_009b_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1168345458774\">No point on the boundary lines is included in the solution as both lines are dashed.<\/p>\n<p id=\"fs-id1168345284829\">The solution is the region that is shaded twice, which is also the solution to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-864074ac6c923db53d28749a677e43cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#50;&#121;&#60;&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" style=\"vertical-align: -4px;\" \/>.<\/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 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345241699\" data-type=\"problem\">\n<p>Solve the system by graphing. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-face6a9184682237c8b3bbee4576e2ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#51;&#120;&#43;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#43;&#121;&#92;&#103;&#101;&#32;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"130\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div id=\"fs-id1168345443262\" data-type=\"solution\">\n<details>\n<summary class=\"answer\">Show answer<\/summary>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-aef8e360ebb3742190eaa73e179b6d58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#103;&#101;&#32;&#51;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<p><span id=\"fs-id1168345270107\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to 3x + 1 and -3x + y is greater than or equal to -4. The area to the left of each line is shaded with the overlapping area shaded a slightly different color.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_021_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to 3x + 1 and -3x + y is greater than or equal to -4. The area to the left of each line is shaded with the overlapping area shaded a slightly different color.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345251566\" class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Solve Applications of Systems of Inequalities<\/h1>\n<p id=\"fs-id1168345560133\">The first thing we\u2019ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system as we did above to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so their graphs will only show Quadrant I.<\/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-id1168345273545\" data-type=\"problem\">\n<p id=\"fs-id1168345384897\">Christy sells her photographs at a booth at a street fair. At the start of the day, she wants to have at least 25 photos to display at her booth. Each small photo she displays costs her ?4 and each large photo costs her ?10. She doesn\u2019t want to spend more than ?200 on photos to display.<\/p>\n<p id=\"fs-id1169749520125\"><span class=\"token\">a)<\/span>\u00a0Write a system of inequalities to model this situation.<\/p>\n<p id=\"fs-id1169754128995\"><span class=\"token\">b)<\/span>\u00a0Graph the system.<\/p>\n<p id=\"fs-id1169752186836\"><span class=\"token\">c)<\/span>\u00a0Could she display 15 small and 5 large photos?<\/p>\n<p id=\"fs-id1169754046303\"><span class=\"token\">d)<\/span>\u00a0Could she display 3 large and 22 small photos?<\/p>\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"title\"><\/div>\n<div data-type=\"title\"><strong>Solution<\/strong><\/div>\n<p>&nbsp;<\/p>\n<p>a) Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-e7bbcde7229c9d7d6f7f2b6793961e97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"28\" style=\"vertical-align: 0px;\" \/> the number of small photos.<span data-type=\"newline\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d93a23224afdb887bb384715060309dc_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;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: -4px;\" \/> the number of large photos<span data-type=\"newline\"><br \/>\n<\/span>To find the system of inequalities, translate the information.<span data-type=\"newline\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-648bf3a88501a14c4f428587b88cf66a_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;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#104;&#101;&#32;&#119;&#97;&#110;&#116;&#115;&#32;&#116;&#111;&#32;&#104;&#97;&#118;&#101;&#32;&#97;&#116;&#32;&#108;&#101;&#97;&#115;&#116;&#32;&#50;&#53;&#32;&#112;&#104;&#111;&#116;&#111;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#115;&#109;&#97;&#108;&#108;&#32;&#112;&#108;&#117;&#115;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#32;&#111;&#102;&#32;&#108;&#97;&#114;&#103;&#101;&#32;&#115;&#104;&#111;&#117;&#108;&#100;&#32;&#98;&#101;&#32;&#97;&#116;&#32;&#108;&#101;&#97;&#115;&#116;&#32;&#50;&#53;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#43;&#121;&#92;&#103;&#101;&#32;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#92;&#36;&#52;&#32;&#102;&#111;&#114;&#32;&#101;&#97;&#99;&#104;&#32;&#115;&#109;&#97;&#108;&#108;&#32;&#97;&#110;&#100;&#32;&#92;&#36;&#49;&#48;&#32;&#102;&#111;&#114;&#32;&#101;&#97;&#99;&#104;&#32;&#108;&#97;&#114;&#103;&#101;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#110;&#111;&#32;&#109;&#111;&#114;&#101;&#32;&#116;&#104;&#97;&#110;&#32;&#92;&#36;&#50;&#48;&#48;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#120;&#43;&#49;&#48;&#121;&#92;&#108;&#101;&#32;&#50;&#48;&#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=\"105\" width=\"529\" style=\"vertical-align: -48px;\" \/><span data-type=\"newline\"><br \/>\n<\/span>We have our system of inequalities. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d350fd27d65945a8f0daf727b6face02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#92;&#103;&#101;&#32;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#49;&#48;&#121;&#92;&#108;&#101;&#32;&#50;&#48;&#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=\"43\" width=\"138\" style=\"vertical-align: -17px;\" \/><\/p>\n<p>b)<\/p>\n<table id=\"fs-id1167834387646\" class=\"unnumbered unstyled can-break\" summary=\"This figure has words and a graph. The words say, \u201cTo graph x + y is greater than or equal 25, graph x + y = 25 as a solid line. Choose (0, 0) as a test point. Since it does not make the inequality true, shade (red) the side that does not include the point (0, 0). To graph 4x + 10y is less than or equal to 200, graph 4x + 10y = 200 as a solid line. Choose (0, 0) as a test point. Since it does make the inequality true, shade (blue) the side that include the point (0, 0). The graph show two line graphed on an x y-coordinte plane. One line has the area to the right shaded. The other line has the area to the left shaded. There is an area where the shaded areas overlap.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">To graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-4c190489f6e3afaa33589133f8bcb75e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#121;&#92;&#103;&#101;&#32;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\" \/>, graph\u00a0<em data-effect=\"italics\">x<\/em>\u00a0+\u00a0<em data-effect=\"italics\">y<\/em>\u00a0= 25 as a solid line.<span data-type=\"newline\"><br \/>\n<\/span>Choose (0, 0) as a test point. Since it does not make the inequality<span data-type=\"newline\"><br \/>\n<\/span>true, shade the side that does not include the point (0, 0) red.<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>To graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1156a84d713fe7a0c6eca16e82141044_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#120;&#43;&#49;&#48;&#121;&#92;&#108;&#101;&#32;&#50;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"118\" style=\"vertical-align: -4px;\" \/>, graph 4<em data-effect=\"italics\">x<\/em>\u00a0+ 10<em data-effect=\"italics\">y<\/em>\u00a0= 200 as a solid line.<span data-type=\"newline\"><br \/>\n<\/span>Choose (0, 0) as a test point. Since it does not make the inequality<span data-type=\"newline\"><br \/>\n<\/span>true, shade the side that includes the point (0, 0) blue.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167835326332\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_010a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span data-type=\"newline\"><br \/>\n<\/span>The solution of the system is the region of the graph that is double shaded and so is shaded darker.<\/p>\n<p>c) To determine if 10 small and 20 large photos would work, we see if the point (10, 20) is in the solution region. It is not. Christy would not display 10 small and 20 large photos.<\/p>\n<p>d) To determine if 20 small and 10 large photos would work, we see if the point (20, 10) is in the solution region. It is. Christy could choose to display 20 small and 10 large photos.<\/p>\n<p id=\"fs-id1168345418460\">Notice that we could also test the possible solutions by substituting the values into each inequality.<\/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 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345329549\" data-type=\"note\">\n<div id=\"fs-id1168345269797\" data-type=\"exercise\">\n<div id=\"fs-id1168345406421\" data-type=\"problem\">\n<p>Mary needs to purchase supplies of answer sheets and pencils for a standardized test to be given to the juniors at her high school. The number of the answer sheets needed is at least 5 more than the number of pencils. The pencils cost &#36;2 and the answer sheets cost &#36;1. Mary\u2019s budget for these supplies allows for a maximum cost of &#36;400.<\/p>\n<p id=\"fs-id1163872103544\">a) Write a system of inequalities to model this situation.<br \/>\nb) Graph the system.<br \/>\nc) Could Mary purchase 100 pencils and 100 answer sheets?<br \/>\nd) Could Mary purchase 150 pencils and 150 answer sheets?<\/p>\n<\/div>\n<div id=\"fs-id1168345291918\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p>a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-64e05514c6708d7fcbffef2b417b8771_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#97;&#92;&#103;&#101;&#32;&#112;&#43;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#43;&#50;&#112;&#92;&#108;&#101;&#32;&#52;&#48;&#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=\"43\" width=\"119\" style=\"vertical-align: -17px;\" \/><\/p>\n<p>b)<br \/>\n<span id=\"fs-id1168345325509\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of a is greater than or equal to p + 5 and a + 2p is less than or equal to 400. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_030_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of a is greater than or equal to p + 5 and a + 2p is less than or equal to 400. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p>c) no<\/p>\n<p>d) no<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\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-id1168345447989\" data-type=\"problem\">\n<p id=\"fs-id1168341862306\">Omar needs to eat at least 800 calories before going to his team practice. All he wants is hamburgers and cookies, and he doesn\u2019t want to spend more than &#36;5. At the hamburger restaurant near his college, each hamburger has 240 calories and costs &#36;1.40. Each cookie has 160 calories and costs &#36;0.50.<\/p>\n<p id=\"fs-id1163872506375\"><span class=\"token\">a)<\/span>\u00a0Write a system of inequalities to model this situation.<span data-type=\"newline\"><br \/>\n<\/span><span class=\"token\">b)<\/span>\u00a0Graph the system.<span data-type=\"newline\"><br \/>\n<\/span><span class=\"token\">c)<\/span>\u00a0Could he eat 3 hamburgers and 1 cookie?<span data-type=\"newline\"><br \/>\n<\/span><span class=\"token\">d)<\/span>\u00a0Could he eat 2 hamburgers and 4 cookies?<\/p>\n<\/div>\n<div id=\"fs-id1168345420640\" 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-id1163871975512\"><span class=\"token\">a)<\/span>\u00a0Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-62bcd55cff5bacf6b29f3badcbd52342_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"28\" style=\"vertical-align: 0px;\" \/> the number of hamburgers.<span data-type=\"newline\"><br \/>\n<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-00315bab48bb50e7946688a49d76153d_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;&#50;&#46;&#53;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#99;&#61;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"26\" style=\"vertical-align: 0px;\" \/> the number of cookies<span data-type=\"newline\"><br \/>\n<\/span>To find the system of inequalities, translate the information.<span data-type=\"newline\"><br \/>\n<\/span>The calories from hamburgers at 240 calories each, plus the calories from cookies at 160 calories each must be more that 800.<\/p>\n<div id=\"fs-id1168345415068\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-35edd613ee2754e748480ebfe187e647_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#48;&#104;&#43;&#49;&#54;&#48;&#99;&#92;&#103;&#101;&#32;&#56;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"143\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"fs-id1168034819709\">The amount spent on hamburgers at &#36;1.40 each, plus the amount spent on cookies at &#36;0.50 each must be no more than &#36;5.00.<\/p>\n<div id=\"fs-id1168345398825\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-eefdcd7145589ffd12824d6f287743dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#52;&#48;&#104;&#43;&#48;&#46;&#53;&#48;&#99;&#92;&#108;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"133\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"fs-id1168034819926\">We have our system of inequalities. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f83b6624c461b424fa23b3146455cee8_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;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#52;&#48;&#104;&#43;&#49;&#54;&#48;&#99;&#92;&#103;&#101;&#32;&#56;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#49;&#46;&#52;&#48;&#104;&#43;&#48;&#46;&#53;&#48;&#99;&#92;&#108;&#101;&#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=\"43\" width=\"163\" style=\"vertical-align: -17px;\" \/><\/p>\n<p id=\"fs-id1163872748969\"><span class=\"token\">b)<\/span><span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<table id=\"fs-id1167835489162\" class=\"unnumbered unstyled can-break\" summary=\"This figure has words and a graph. The words say, \u201cTo graph 240h + 160c is greater than or equal to 800, graph 240h + 160c = 800 as solid line. Choose (0, 0) as a test point. Since it does not make the inequality true, shade (red) the side that does not include the point (0, 0).\u201d The graph shows two solide lines. The area to the right of one line is shaded, and the area to the left of the other line is shaded. There is an area where to the two shaded areas overlap.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\">To graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-35edd613ee2754e748480ebfe187e647_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#48;&#104;&#43;&#49;&#54;&#48;&#99;&#92;&#103;&#101;&#32;&#56;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"143\" style=\"vertical-align: -3px;\" \/> graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-de00912023c4953f1c00dafec004952d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#48;&#104;&#43;&#49;&#54;&#48;&#99;&#61;&#56;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"143\" style=\"vertical-align: -2px;\" \/> as a solid line.<span data-type=\"newline\"><br \/>\n<\/span>Choose (0, 0) as a test point. it does not make the inequality true.<span data-type=\"newline\"><br \/>\n<\/span>So, shade (red) the side that does not include the point (0, 0).<span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span><span data-type=\"newline\"><br \/>\n<\/span>To graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-eefdcd7145589ffd12824d6f287743dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#52;&#48;&#104;&#43;&#48;&#46;&#53;&#48;&#99;&#92;&#108;&#101;&#32;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"133\" style=\"vertical-align: -3px;\" \/>, graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-66841686dbbb1a0c26fd3342ee57615c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#52;&#48;&#104;&#43;&#48;&#46;&#53;&#48;&#99;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"133\" style=\"vertical-align: -2px;\" \/> as a solid line.<span data-type=\"newline\"><br \/>\n<\/span>Choose (0,0) as a test point. It makes the inequality true. So, shade<span data-type=\"newline\"><br \/>\n<\/span>(blue) the side that includes the point.<\/td>\n<td data-valign=\"top\"><span id=\"fs-id1167834376306\" data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_011a_img_new.jpg\" alt=\".\" data-media-type=\"image\/png\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span data-type=\"newline\"><br \/>\n<\/span>The solution of the system is the region of the graph that is double shaded and so is shaded darker.<\/p>\n<p id=\"fs-id1163872841697\"><span class=\"token\">c)<\/span>\u00a0To determine if 3 hamburgers and 2 cookies would meet Omar\u2019s criteria, we see if the point (3, 1) is in the solution region. It is. He might choose to eat 3 hamburgers and 2 cookies.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<p><span class=\"token\">d)<\/span>\u00a0To determine if 2 hamburgers and 4 cookies would meet Omar\u2019s criteria, we see if the point (2, 4) is in the solution region. It is. He might choose to eat 2 hamburgers and 4 cookies.<\/p>\n<p id=\"fs-id1168345743092\">We could also test the possible solutions by substituting the values into each inequality.<\/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 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"fs-id1168345256115\" data-type=\"problem\">\n<p id=\"fs-id1168345376826\">Tension needs to eat at least an extra 1,000 calories a day to prepare for running a marathon. He has only &#36;25 to spend on the extra food he needs and will spend it on &#36;0.75 donuts which have 360 calories each and &#36;2 energy drinks which have 110 calories.<\/p>\n<p id=\"fs-id1167835166563\">a) Write a system of inequalities that models this situation.<br \/>\nb) Graph the system.<br \/>\nc) Can he buy 8 donuts and 4 energy drinks?<br \/>\nd) Can he buy 1 donut and 3 energy drinks?<\/p>\n<\/div>\n<div id=\"fs-id1168345229342\" data-type=\"solution\">\n<details open=\"open\">\n<summary class=\"answer\">Show answer<\/summary>\n<p>a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-1ef5c013f8b6a656f1767fd284946ab8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#48;&#46;&#55;&#53;&#100;&#43;&#50;&#101;&#92;&#108;&#101;&#32;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#54;&#48;&#100;&#43;&#49;&#49;&#48;&#101;&#92;&#103;&#101;&#32;&#49;&#48;&#48;&#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=\"43\" width=\"171\" style=\"vertical-align: -17px;\" \/><\/p>\n<p>b)<br \/>\n<span id=\"fs-id1168345250261\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 0.75d + 2e is less than or equal to 25 and 360d + 110e is greater than or equal to 1000. The area to the left or right of each line is shaded slightly different colors with the overlapping area also shaded a slightly different color.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure05_06_025_img.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 0.75d + 2e is less than or equal to 25 and 360d + 110e is greater than or equal to 1000. The area to the left or right of each line is shaded slightly different colors with the overlapping area also shaded a slightly different color.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p>c) yes<\/p>\n<p>d) no<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345367666\" class=\"media-2\" data-type=\"note\">\n<p id=\"fs-id1168345274887\">Access these online resources for additional instruction and practice with graphing systems of linear inequalities.<\/p>\n<ul id=\"fs-id1168345441091\" data-display=\"block\">\n<li><a href=\"http:\/\/www.openstax.org\/l\/25GSInequal1\">Graphical System of Inequalities<\/a><\/li>\n<li><a href=\"http:\/\/www.openstax.org\/l\/25GSInequal2\">Systems of Inequalities<\/a><\/li>\n<li><a href=\"http:\/\/www.openstax.org\/l\/25GSInequal3\">Solving Systems of Linear Inequalities by Graphing<\/a><\/li>\n<\/ul>\n<h1 data-type=\"title\">Key Concepts<\/h1>\n<ul id=\"fs-id1168345434983\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">To Solve a System of Linear Inequalities by Graphing<\/strong>\n<ol id=\"fs-id1169754231290\" class=\"stepwise\" type=\"1\">\n<li>Graph the first inequality.\n<ul id=\"fs-id1171792500552\" data-bullet-style=\"circled\">\n<li>Graph the boundary line.<\/li>\n<li>Shade in the side of the boundary line where the inequality is true.<\/li>\n<\/ul>\n<\/li>\n<li>On the same grid, graph the second inequality.\n<ul id=\"fs-id1171792759774\" data-bullet-style=\"circled\">\n<li>Graph the boundary line.<\/li>\n<li>Shade in the side of that boundary line where the inequality is true.<\/li>\n<\/ul>\n<\/li>\n<li>The solution is the region where the shading overlaps.<\/li>\n<li>Check by choosing a test point.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<div id=\"fs-id1168345271071\" class=\"practice-perfect\" data-depth=\"2\">\n<h1 data-type=\"title\">4.5 Exercise Set<\/h1>\n<p id=\"fs-id1169753987508\">In the following exercises, determine whether each ordered pair is a solution to the system.<\/p>\n<ol class=\"twocolumn\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1407\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-1-300x122.png\" alt=\"\" width=\"138\" height=\"56\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-1-300x122.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-1-768x312.png 768w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-1-65x26.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-1-225x91.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-1-350x142.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-1.png 838w\" sizes=\"auto, (max-width: 138px) 100vw, 138px\" \/>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f054d79e93ab302f68adcad7d4c3ec21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f63e6ea0c8474555192b305e3472481d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1408\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-2-300x128.png\" alt=\"\" width=\"141\" height=\"60\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-2-300x128.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-2-65x28.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-2-225x96.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-2-350x149.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-2.png 766w\" sizes=\"auto, (max-width: 141px) 100vw, 141px\" \/>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-eb87bfe9ba69a89c4ac12d88180c12e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-28a186f2424d2e935d6aa6388441b6d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1409\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-3-300x115.png\" alt=\"\" width=\"141\" height=\"54\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-3-300x115.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-3-768x294.png 768w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-3-65x25.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-3-225x86.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-3-350x134.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-3.png 878w\" sizes=\"auto, (max-width: 141px) 100vw, 141px\" \/>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-3fca1a384cf5042876a719066cbbb127_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-87f21cb31fb244b94346350d581a367b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ol>\n<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ec968f0526c060a898530ff5aebd9877_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#51;&#121;&#92;&#103;&#101;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#45;&#54;&#121;&#60;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"124\" style=\"vertical-align: -17px;\" \/>\n<ol type=\"a\">\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6a248acf47140210663edaf2e4d5eb76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"42\" style=\"vertical-align: -7px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-cc7ee02701c0ea338e43c64eee5af2a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"42\" style=\"vertical-align: -7px;\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p id=\"fs-id1169749956485\">In the following exercises, solve each system by graphing.<\/p>\n<ol class=\"twocolumn\" start=\"5\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1410\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-5-300x144.png\" alt=\"\" width=\"140\" height=\"67\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-5-300x144.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-5-65x31.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-5-225x108.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-5-350x169.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-5.png 756w\" sizes=\"auto, (max-width: 140px) 100vw, 140px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-0ac0e3186f612203ef3229505f40a6f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#60;&#50;&#120;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#108;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1411\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-7-300x117.png\" alt=\"\" width=\"139\" height=\"54\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-7-300x117.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-7-768x300.png 768w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-7-65x25.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-7-225x88.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-7-350x137.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-7.png 866w\" sizes=\"auto, (max-width: 139px) 100vw, 139px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-d319f3c31b83cb772922b8c5aff23cf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#121;&#92;&#108;&#101;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#103;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-47e960df8c74191ede88b27a3e741143_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#45;&#53;&#121;&#60;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#52;&#121;&#92;&#103;&#101;&#32;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1412\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-10-300x112.png\" alt=\"\" width=\"140\" height=\"52\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-10-300x112.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-10-768x287.png 768w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-10-65x24.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-10-225x84.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-10-350x131.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-10.png 920w\" sizes=\"auto, (max-width: 140px) 100vw, 140px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-c8482f4bfc902bb065c3b610da8cde58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#50;&#121;&#60;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#108;&#101;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"102\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-8a360d3c3c09797ea6614b0db16e5542_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#120;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#92;&#108;&#101;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"118\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f63ab4c635e95aa90bf7865225bf6012_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#103;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#60;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"103\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-736e3c16a5aa9022b03c41895b263c23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#52;&#121;&#60;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#60;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"111\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-b4718a8b8d6746229bfbc0991ec716d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#92;&#103;&#101;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#92;&#108;&#101;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"62\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1413\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-15.png\" alt=\"\" width=\"139\" height=\"54\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-15.png 864w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-15-300x117.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-15-768x299.png 768w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-15-65x25.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-15-225x88.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-15-350x136.png 350w\" sizes=\"auto, (max-width: 139px) 100vw, 139px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1414\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-17-300x109.png\" alt=\"\" width=\"140\" height=\"51\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-17-300x109.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-17-768x279.png 768w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-17-65x24.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-17-225x82.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-17-350x127.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-17.png 920w\" sizes=\"auto, (max-width: 140px) 100vw, 140px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1415\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-18-300x118.png\" alt=\"\" width=\"140\" height=\"55\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-18-300x118.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-18-768x302.png 768w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-18-65x26.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-18-225x88.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-18-350x137.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-18.png 866w\" sizes=\"auto, (max-width: 140px) 100vw, 140px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-6f70c02d3b1f54e5e5119ab80cfcb331_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#121;&#92;&#108;&#101;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#52;&#121;&#60;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"117\" style=\"vertical-align: -17px;\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1416\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-20-300x110.png\" alt=\"\" width=\"141\" height=\"52\" srcset=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-20-300x110.png 300w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-20-768x282.png 768w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-20-65x24.png 65w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-20-225x83.png 225w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-20-350x129.png 350w, https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/uploads\/sites\/361\/2021\/08\/4.5-20.png 914w\" sizes=\"auto, (max-width: 141px) 100vw, 141px\" \/><\/li>\n<\/ol>\n<p id=\"fs-id1169751892445\">In the following exercises, translate to a system of inequalities and solve.<\/p>\n<ol start=\"21\">\n<li>Caitlyn sells her drawings at the county fair. She wants to sell at least 60 drawings and has portraits and landscapes. She sells the portraits for &#36;15 and the landscapes for &#36;10. She needs to sell at least &#36;800 worth of drawings in order to earn a profit.\n<ol type=\"a\">\n<li>Write a system of inequalities to model this situation.<\/li>\n<li>Graph the system.<\/li>\n<li>Will she make a profit if she sells 20 portraits and 35 landscapes?<\/li>\n<li>Will she make a profit if she sells 50 portraits and 20 landscapes?<\/li>\n<\/ol>\n<\/li>\n<li>Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than &#36;500. The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is &#36;3 and for a package the cost is &#36;7.\n<ol type=\"a\">\n<li>Write a system of inequalities to model this situation.<\/li>\n<li>Graph the system.<\/li>\n<li>Can she mail 60 cards and 26 packages?<\/li>\n<li>Can she mail 90 cards and 40 packages?<\/li>\n<\/ol>\n<\/li>\n<li>Jocelyn is pregnant and needs to eat at least 500 more calories a day than usual. When buying groceries one day with a budget of &#36;15 for the extra food, she buys bananas that have 90 calories each and chocolate granola bars that have 150 calories each. The bananas cost &#36;0.35 each and the granola bars cost &#36;2.50 each.\n<ol type=\"a\">\n<li>Write a system of inequalities to model this situation.<\/li>\n<li>Graph the system.<\/li>\n<li>Could she buy 5 bananas and 6 granola bars?<\/li>\n<li>Could she buy 3 bananas and 4 granola bars?<\/li>\n<\/ol>\n<\/li>\n<li>Jocelyn desires to increase both her protein consumption and caloric intake. She desires to have at least 35 more grams of protein each day and no more than an additional 200 calories daily. An ounce of cheddar cheese has 7 grams of protein and 110 calories. An ounce of parmesan cheese has 11 grams of protein and 22 calories.\n<ol type=\"a\">\n<li>Write a system of inequalities to model this situation.<\/li>\n<li>Graph the system.<\/li>\n<li>Could she eat 1 ounce of cheddar cheese and 3 ounces of parmesan cheese?<\/li>\n<li>Could she eat 2 ounces of cheddar cheese and 1 ounce of parmesan cheese?<\/li>\n<\/ol>\n<\/li>\n<li>Tickets for an American Baseball League game for 3 adults and 3 children cost less than ?75, while tickets for 2 adults and 4 children cost less than &#36;62.\n<ol type=\"a\">\n<li>Write a system of inequalities to model this problem.<\/li>\n<li>Graph the system.<\/li>\n<li>Could the tickets cost &#36;20 for adults and &#36;8 for children?<\/li>\n<li>Could the tickets cost &#36;15 for adults and &#36;5 for children?<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\">\n<h1 id=\"fs-id1169752082080\">Answers:<\/h1>\n<ol class=\"twocolumn\">\n<li style=\"list-style-type: none;\">\n<ol type=\"a\">\n<li>true<\/li>\n<li>false<\/li>\n<\/ol>\n<ol type=\"a\">\n<li>false<\/li>\n<li>true<\/li>\n<\/ol>\n<ol type=\"a\">\n<li>true<\/li>\n<li>false<\/li>\n<\/ol>\n<ol type=\"a\">\n<li>true<\/li>\n<li>true<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345276530\" data-type=\"problem\">\n<p>5.<\/p>\n<\/div>\n<div id=\"fs-id1168345273986\" data-type=\"solution\"><span id=\"fs-id1168345708305\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is less than or equal to 3x + 2 and y is greater than x \u2013 1. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_201_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is less than or equal to 3x + 2 and y is greater than x \u2013 1. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168341851439\" data-type=\"problem\">\n<p>6.<\/p>\n<\/div>\n<div id=\"fs-id1168345634904\" data-type=\"solution\"><span id=\"fs-id1168345623920\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is less than 2x - 1 and y is less than or equal to -(1\/2)x + 4. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_203_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is less than 2x - 1 and y is less than or equal to -(1\/2)x + 4. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345462056\" data-type=\"exercise\">\n<div id=\"fs-id1168345301976\" data-type=\"problem\">\n<p id=\"fs-id1168345535241\">7.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1168345418364\" data-type=\"exercise\">\n<div id=\"fs-id1168341917954\" data-type=\"solution\"><span id=\"fs-id1168345434088\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of x \u2013 y is greater than 1 and y is less than \u2013(1\/4)x + 3. The area to the right or below each line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_205_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of x \u2013 y is greater than 1 and y is less than \u2013(1\/4)x + 3. The area to the right or below each line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345274250\" data-type=\"problem\">\n<p>8.<\/p>\n<\/div>\n<div id=\"fs-id1168345346375\" data-type=\"solution\"><span id=\"fs-id1168345647738\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 3x \u2013 y is less than or equal to 6 and y is greater than or equal to \u2013(1\/2)x. The area to the right or above each line is shaded different colors with the overlapping area also shaded a different color.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_207_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 3x \u2013 y is less than or equal to 6 and y is greater than or equal to \u2013(1\/2)x. The area to the right or above each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345432909\" data-type=\"problem\">\n<p>9.<\/p>\n<\/div>\n<div id=\"fs-id1168345540531\" data-type=\"solution\"><span id=\"fs-id1168345560367\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 2x \u2013 5y is less than 10 and 3x +4y is greater than or equal to 12. The area to the right above each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_209_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 2x \u2013 5y is less than 10 and 3x +4y is greater than or equal to 12. The area to the right above each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345254775\" data-type=\"problem\">\n<p>10.<\/p>\n<\/div>\n<div id=\"fs-id1168341960737\" data-type=\"solution\"><span id=\"fs-id1168341852659\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 2x + 2y is greater than -4 and \u2013x + 3y is greater than or equal to 9. The area to the right or above each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_211_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 2x + 2y is greater than -4 and \u2013x + 3y is greater than or equal to 9. The area to the right or above each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345217910\" data-type=\"problem\">\n<p>11.<\/p>\n<\/div>\n<div id=\"fs-id1168345744963\" data-type=\"solution\"><span id=\"fs-id1168345511763\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of x \u2013 2y is less than 3 and y is less than or equal to 1. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_213_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of x \u2013 2y is less than 3 and y is less than or equal to 1. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345633331\" data-type=\"problem\">\n<p>12.<\/p>\n<\/div>\n<div id=\"fs-id1168345644626\" data-type=\"solution\"><span id=\"fs-id1168345292038\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to (-1\/2)x - 3 and x is less than or equal to 2. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_215_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to (-1\/2)x - 3 and x is less than or equal to 2. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345540621\" data-type=\"problem\">\n<p>13.<\/p>\n<\/div>\n<div id=\"fs-id1168345387414\" data-type=\"solution\"><span id=\"fs-id1168341917038\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to (3\/4)x - 2 and y is less than 2. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_217_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to (3\/4)x - 2 and y is less than 2. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color. One line is dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168345387655\" data-type=\"problem\">\n<p>14.<\/p>\n<\/div>\n<div id=\"fs-id1168341864160\" data-type=\"solution\"><span data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 3x \u2013 4y is less than 8 and x is less than 1. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_219_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 3x \u2013 4y is less than 8 and x is less than 1. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div data-type=\"problem\">\n<p>15.<\/p>\n<\/div>\n<div id=\"fs-id1168345738042\" data-type=\"solution\"><span id=\"fs-id1168345214653\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of x is greater than or equal to 3 and y less than or equal to 2. The area to the right or below each line is shaded different colors with the overlapping area also shaded a different color.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_221_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of x is greater than or equal to 3 and y less than or equal to 2. The area to the right or below each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<p id=\"fs-id1168345439795\">16. No solution<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<p><span id=\"fs-id1168345644259\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 2x + 4y is greater than 4 and y is less than or equal to (-1\/2)x - 2. The area to the left or right of each line is shaded different colors. There is no area where the shaded areas overlap. One line is dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_223_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 2x + 4y is greater than 4 and y is less than or equal to (-1\/2)x - 2. The area to the left or right of each line is shaded different colors. There is no area where the shaded areas overlap. One line is dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">17. No solution<span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1168345434496\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of -2x + 6y is less than 0 and 6y is greater than 2x + 4. The area to the left or right of each line is shaded different colors. There is no area where the shaded areas overlap. Both lines are dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_225_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of -2x + 6y is less than 0 and 6y is greater than 2x + 4. The area to the left or right of each line is shaded different colors. There is no area where the shaded areas overlap. Both lines are dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"fs-id1168341958605\" data-type=\"problem\">\n<p>18.<\/p>\n<\/div>\n<div id=\"fs-id1168345429012\" data-type=\"solution\"><span id=\"fs-id1168345398337\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to -3x + 2 and 3x + y is greater than 5. The area to the right of each line is shaded different colors. One line is within the shaded area of the other. One line is dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_227_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is greater than or equal to -3x + 2 and 3x + y is greater than 5. The area to the right of each line is shaded different colors. One line is within the shaded area of the other. One line is dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<p id=\"fs-id1168345560336\">19.<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-eba01ee32ad895e62b57f624de2d0d39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#43;&#52;&#121;&#60;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<p><span data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of y is less than or equal to (negative 1\/4)x \u2013 2 and x + 4y is less than 6. The area below each line is shaded different colors. One line is within the shaded area of the other. One line is dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_229_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of y is less than or equal to (negative 1\/4)x \u2013 2 and x + 4y is less than 6. The area below each line is shaded different colors. One line is within the shaded area of the other. One line is dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<p id=\"fs-id1168345638227\">20. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-ce0202e1bdf98d4a988b4343b5cde076_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#120;&#43;&#54;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"71\" style=\"vertical-align: -4px;\" \/> &gt; <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;\" \/><span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<p><span id=\"fs-id1168345560952\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 3y is greater than x + 2 and -2x + 6y is greater than 8. The area above each line is shaded different colors. One line is within the shaded area of the other. Both lines are dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_231_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 3y is greater than x + 2 and -2x + 6y is greater than 8. The area above each line is shaded different colors. One line is within the shaded area of the other. Both lines are dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/td>\n<td style=\"width: 33.3333%;\"><span class=\"token\">21. a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-12ed782f70d889859906614695ede14d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#112;&#43;&#108;&#92;&#103;&#101;&#32;&#54;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#49;&#53;&#112;&#43;&#49;&#48;&#108;&#92;&#103;&#101;&#32;&#56;&#48;&#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=\"43\" width=\"142\" style=\"vertical-align: -17px;\" \/><\/p>\n<p><span class=\"token\">b)<\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1168345692981\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of p + l is greater than or equal to 60 and 15p + 10l is greater than or equal to 800. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_233_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of p + l is greater than or equal to 60 and 15p + 10l is greater than or equal to 800. The area to the left of each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p><span class=\"token\">c)<\/span> No<\/p>\n<p><span class=\"token\">d)<\/span> Yes<\/td>\n<td style=\"width: 33.3333%;\"><span class=\"token\">22. a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-bf0fa5657118d0f0ff5bf07ce1b0e2b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#55;&#112;&#43;&#51;&#99;&#92;&#108;&#101;&#32;&#53;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#112;&#92;&#103;&#101;&#32;&#50;&#99;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"126\" style=\"vertical-align: -17px;\" \/><\/p>\n<p><span class=\"token\">b)<\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1168345529466\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 7p + 3c is less than or equal to 500 and p is greater than or equal to 2c + 4. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_235_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 7p + 3c is less than or equal to 500 and p is greater than or equal to 2c + 4. The area to the left or below each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p><span class=\"token\">c)<\/span> Yes<\/p>\n<p><span class=\"token\">d)<\/span> No<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\"><span class=\"token\">23. a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-5c5fda81a5b595db3ccec2201a6ebd53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#57;&#48;&#98;&#43;&#49;&#53;&#48;&#103;&#92;&#103;&#101;&#32;&#53;&#48;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#48;&#46;&#51;&#53;&#98;&#43;&#50;&#46;&#53;&#48;&#103;&#92;&#108;&#101;&#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=\"43\" width=\"161\" style=\"vertical-align: -17px;\" \/><\/p>\n<p><span data-type=\"newline\">b)<br \/>\n<\/span><span id=\"fs-id1168345287674\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 90b + 150g is greater than or equal to 500 and 0.35b + 2.50g is less than or equal to 15. The area to the right or below each line is shaded different colors with the overlapping area also shaded a different color.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_237_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 90b + 150g is greater than or equal to 500 and 0.35b + 2.50g is less than or equal to 15. The area to the right or below each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p><span class=\"token\">c)<\/span> No<\/p>\n<p><span class=\"token\">d)<\/span> Yes<\/td>\n<td style=\"width: 33.3333%;\"><span class=\"token\">24. a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-f1d9439065375ac9817cdc61272386f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#55;&#99;&#43;&#49;&#49;&#112;&#92;&#103;&#101;&#32;&#51;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#49;&#49;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#43;&#50;&#50;&#112;&#92;&#108;&#101;&#32;&#50;&#48;&#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=\"43\" width=\"153\" style=\"vertical-align: -17px;\" \/><\/p>\n<p><span class=\"token\">b)<\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1168345448185\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 7c + 11p is greater than or equal to 35 and 110c + 22p is less than or equal to 200. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_239_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 7c + 11p is greater than or equal to 35 and 110c + 22p is less than or equal to 200. The area to the left or right of each line is shaded different colors with the overlapping area also shaded a different color.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p><span class=\"token\">c)<\/span> Yes<\/p>\n<p><span class=\"token\">d)<\/span> No<\/td>\n<td style=\"width: 33.3333%;\"><span class=\"token\">25. a) <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-content\/ql-cache\/quicklatex.com-9c26b7cbec22b65f5ba212a35baa1303_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#97;&#43;&#51;&#99;&#60;&#55;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#125;&#43;&#52;&#99;&#60;&#54;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<p><span class=\"token\">b)<\/span><span data-type=\"newline\"><br \/>\n<\/span><span id=\"fs-id1168341919046\" data-type=\"media\" data-alt=\"This figure shows a graph on an x y-coordinate plane of 3a + 3c is less than 75 and 2a + 4c is less than 62. The area to the left ofeach line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\"><img decoding=\"async\" src=\"https:\/\/opentextbc.ca\/oerdiscipline\/wp-content\/uploads\/sites\/361\/2021\/08\/CNX_ElemAlg_Figure_05_06_241_img_new.jpg\" alt=\"This figure shows a graph on an x y-coordinate plane of 3a + 3c is less than 75 and 2a + 4c is less than 62. The area to the left ofeach line is shaded different colors with the overlapping area also shaded a different color. Both lines are dotted.\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<p><span class=\"token\">c)<\/span> No<\/p>\n<p><span class=\"token\">d)<\/span> Yes<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":125,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["lynn-marecek","maryanne-anthony-smith"],"pb_section_license":""},"chapter-type":[],"contributor":[63,64],"license":[],"class_list":["post-1438","chapter","type-chapter","status-publish","hentry","contributor-lynn-marecek","contributor-maryanne-anthony-smith"],"part":1094,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1438","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":1,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1438\/revisions"}],"predecessor-version":[{"id":1439,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1438\/revisions\/1439"}],"part":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/parts\/1094"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapters\/1438\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/media?parent=1438"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/pressbooks\/v2\/chapter-type?post=1438"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/contributor?post=1438"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/businesstechnicalmath\/wp-json\/wp\/v2\/license?post=1438"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}