Introductory Algebra Copyright © 2021 by Izabela Mazur is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.
© 2021 Izabela Mazur
Introductory Algebra was adapted by Izabela Mazur using content from Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith and from Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which are both under CC BY 4.0 Licences.
The adaptation was done with the goal of bringing content into alignment with the British Columbia Adult Basic Education learning outcomes for Mathematics: Intermediate Level Algebra. These changes and additions are © 2021 by Izabela Mazur and are licensed under a CC BY 4.0 Licence:
- Combined and reorganized chapters from the original two books.
- Adjusted formatting and textboxes so the content displays well in the Pressbooks platform.
- Added, removed, and reorganized questions and examples where appropriate.
- Replaced American examples and spellings with Canadian ones.
See the end of each chapter to see what content was adapted from where.
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Counting Numbers: 1, 2, 3, …
is named as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.
using words.
using words.
.
.
to the digit in the given place value.
to the nearest:
is 5, so 15 is 
.
, we say that 8 and 9 are factors of 72. When we write 
, we say we have factored 72
. Seventy-two has many factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, and 72
, then a and b are factors of m.
is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer!
, the result would still be the same. Finish the prime factorization and verify this for yourself.
and the prime factors of 18 
are included in the LCM 
. So 36 is the least common multiple of 12 and 18
is a multiple of 
, then 
, then 
are factors of 
years old and Alex is 
, so Alex is 
years older than Greg. When Greg was 
, Alex was 
. When Greg is 
, Alex will be 
. No matter what Greg’s age is, Alex’s age will always be 
. Then we could use 
to represent Alex’s age. See the table below.
.
and 
mean 
(three times 
) or 
(three times 
)? To make it clear, use • or parentheses for multiplication.
and 
.
and 
means subtract 
.
and 
means multiply 
.
.
minus 
is called the 
> 
can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,
, it means 
or 
. We read this 
it means not equal.
> 
< 
< 
> 
< 
< 
> 
< 
>. in each expression to compare the fuel economy of the cars.
![\left[\phantom{\rule{0.5em}{0ex}}\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-6366aa9031ead5bd088fb8b907907f10_l3.svg "Rendered by QuickLaTeX.com")
![8\left(14-8\right)\phantom{\rule{4em}{0ex}}21-3\left[2+4\left(9-8\right)\right]\phantom{\rule{4em}{0ex}}24\div \left\{13-2\left[1\left(6-5\right)+4\right]\right\}](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-7598f6833080f5808d041872936979d0_l3.svg "Rendered by QuickLaTeX.com")
we’d first multiply 
to get 
. We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write 
as 
and 
. In expressions such as 
is a factor multiplied by itself 
is read 
power.
and 
, we have special names.
to the second power, or 
, so 
means 
means 
.
.
![\text{Simplify:}\phantom{\rule{0.2em}{0ex}}5+{2}^{3}+3\left[6-3\left(4-2\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-0925a6ac1373608a585730b984c7311e_l3.svg "Rendered by QuickLaTeX.com")
.
![9+{5}^{3}-\left[4\left(9+3\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-0c738327e6a736035e5d49618465c2cc_l3.svg "Rendered by QuickLaTeX.com")
![{7}^{2}-2\left[4\left(5+1\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-840c7c00f2c5c201a98f023f0bb13613_l3.svg "Rendered by QuickLaTeX.com")
.
power.
< 
< 
> 
> 
![2\left[1+3\left(10-2\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-30e1bf8798a5926bd4107059d2e499ad_l3.svg "Rendered by QuickLaTeX.com")
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mountains with an elevation of over 
The table shows the ten tallest. Use this table to fill in the appropriate inequality symbol.
, the expression 
, the expression 
.
means 
means 
, we substitute 
, we substitute 
and in part b) we wrote 
. Both the dot and the parentheses tell us to multiply.
when 
.
.
.
.
.
has a value of 
.
.
.
.
, the expression 
has a value of 
.
means 
and is different from the expression 
, which means 
.
.
.
.
.
is 
. The table below gives the coefficients for each of the terms in the left column.
. Then identify the coefficient of each term.
, and 
is 
is 
and 
and 
both have 
, and constants.
and 
are like terms because they both have 
and 
and 
are like terms because they all have 
has no like terms in the given expression because no other terms contain the two variables 
.
would simplify to? If you thought 
.
.
is in simplest form.
, 
, 
, 
, 
, 
and 
and 
and 
represent the width of the window. Write an expression for the height of the window.
represent the length of the rectangle. Write an expression for the width of the rectangle.
and 
and 
represent the number of classical CDs. Write an expression for the number of rock CDs.
represent the number of female cousins. Write an expression for the number of boy cousins.
represent the number of tens. Write an expression for the number of fives.
times the sum of 
and “the sum of 
. The 
___
d) 
.
a) 
___
___
.
b) 
___
.
means the opposite of the number a.
c) 
.
.
.
b) 
.
, when a) 
b) 
.
, when a) 
b) 
.
b) 11
are opposites because they are the same distance from 0 on the number line. They are both two units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.
.
units away from 
.
units away from 
.
units away from 
on the number line is zero units.
for all numbers
b) 
c) 
.
b) 
c) 
b) 
c) 
for each of the following pairs of numbers:
___
b) 
c) 
___
d) –
___
___ –
c) 
___
b) 
___-
___
d) 
.
.
.
.
b) 
c) 
d) 
.
b) 
c) 
d) 
.
c) 
d) 
b) 
c) 
d) 
.
d) 
and 
.
. Watch for similarities to the last example 
.
.
b) 
.
b) 
.
b) 
.
. We realize this means the sum of 
.
b) 
.
b) 
.
b) 
.
, you really don’t want to have to count out 37 blue counters and 53 red counters. With the model in your mind, can you visualize what you would do to solve the problem?
, there are 16 more red counters.
. Again, imagine 74 red counters and 27 more red counters, so we’d have 101 red counters. This means the sum is 
.
b) 
.
The answer will be negative because there are more negatives than positives.
b) 
.
b)
b) 
.
b)
.
.
.
as 
take away 
When you use counters, you can think of subtraction the same way!
, we restate the problem as 
. Watch for similarities to the last example 
.
take away 
b) 
.
b) 
.
b) 
.
, we restate it as 
. We start with 5 positives. We need to take away 3 negatives, but there are no negatives to take away. So we add neutral pairs until we have 3 negatives to take away.
b) 
.
b) 
.
b) 
.
and 
. You will often see this idea, the subtraction property, written as follows:
.
helps when you are subtracting negative numbers. Make sure that you understand how 
and 
b) 
and 
.
and 
b) 
and 
.
and 
b) 
and 
.
.
and 
b) 
.
and 
b) 
and 
.
and 
b) 
and 
.
.
.
.
when
when
___
![25-\left[10-\left(3-12\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-93bff9c2baf995720da4030aee245f22_l3.svg "Rendered by QuickLaTeX.com")
![32-\left[5-\left(15-20\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-cb825f53789b236f424ae01b5da9a187_l3.svg "Rendered by QuickLaTeX.com")
° Celsius, recorded in the Sahara Desert in 1922. The lowest recorded temperature was 
° below 
sign in algebra? Explain how they differ.
b) 
million
means add a, b times. Here, we are using the model just to help us discover the pattern.
It means subtract 5, 3 times. Looking at subtraction as “taking away,” it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away 5 three times.
b) 
c) 
d) 
.
b) 
c) 
d) 
.
b) 28 c) 
d) 60
b) 
c) 
d) 
.
b) 54 c) 
d) 39
b) 
.
b) 
.
b) 
.
because 
. In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.
b) 
.
b) 
.
b) 
.
.
.
.
b) 
.
.
to the 
power. In part b), the exponent means to raise just the 2 to the 
b) 
.
b) 
.
.
.
.
.
.
.
.
.
.
, evaluate: a) 
b) 
.
, evaluate a) 
b) 
.
, evaluate a) 
b) 
.
when 
and 
.
and 
.
when 
.
when a) 
and b) 
.
when a) 
and b) 
.
when a) 
and b) 
.
when 
.
. Use parentheses to show multiplication.
when 
.
when 
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b)
b)
.
and 
![9-2\left[3-8\left(-2\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-d53067a8da43befde6f153cea9ad046d_l3.svg "Rendered by QuickLaTeX.com")
![11-3\left[7-4\left(-2\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-736f06f6d3a1259461be721afeff8c7a_l3.svg "Rendered by QuickLaTeX.com")
when
when
when 
when 
when 
when 
when
when
when 
when 
when
when
when
when
when
when
when 
when
when
and 
and 
° . That same day, the high temperature in Fort Nelson, British Columbia was 
°, and the high temperature in Whitefield, New Hampshire was 
°. What was the difference between the temperature in Palm Springs and the temperature in Whitefield?
in his checking account. He deposits $225 to the account. What is the new balance?
in his checking account. He deposits $281 to the account. What is the new balance?
°
;5
means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See 
represents two of three equal parts. In the fraction 
and
pieces, or, in other words, one whole pie.
. This leads us to the property of one that tells us that any number, except zero, divided by itself is 1
pieces, or one whole pie. We ate the same amount—one whole pie.
is equivalent to 
. In other words, they are 
How could we take a pizza that is cut into 2 pieces and cut it into 8 pieces? We could cut each of the 2 larger pieces into 4 smaller pieces! The whole pizza would then be cut into 
. See 
.
are numbers where 
, then
are equivalent fractions.
.
are equivalent to 
.
answers may vary
.
answers may vary
is not simplified because 
.
.
.
.
.
.
.
.
.
.
are numbers where 
, then
.
.
.
. So, for example, 
.
.
as a fraction.
.
.
.
. A number and its reciprocal multiply to 1.
is 
, since 
.
.
.
, then
.
.
.
.
.
.
means 
.
.
.
.
.
.
, we first simplify the numerator and the denominator separately. Then we divide.
.
.
.
.
.
.
.
and 
.
, then 
, then 
. To multiply fractions, multiply the numerators and multiply the denominators.
.
Any number, except zero, divided by itself is one.
cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the cookie recipe.
.
answers may vary
answers may vary
cups b) answers will vary
are numbers where 
, then
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
b) 
.
b) 
b) 
b) 
.
b) 
.
.
.
.
.
.
when a) 
b) 
.
when a) 
b) 
.
when a) 
b) 
.
b) 
when 
.
when 
.
when 
.
when 
and 
to make it easy to remove common factors.
when 
and 
.
when 
and 
.
.
when 
.
when 
.
b) 
b) 
b) 
b) 
when
when
when
when
when
when
when 
when 
and 
when 
when 
b) 4
b)
yard
.
.
.
.
.
.
.
to the nearest a) hundredth b) tenth c) whole number.
to the nearest a) thousandth b) hundredth c) tenth.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
is the same as the number of decimal places in 0.374?
or 
. This leads to the following method for converting a fraction to a decimal.
. Now divide.
as a decimal.
as a decimal.
as a decimal.
.
.
.
, where p and q are integers and 
.
are rational numbers. Each numerator and each denominator is an integer.
. So, clearly, some decimals are rational.
, we can write it as an improper fraction, 
. So 7.3 is the ratio of the integers 73 and 10. It is a rational number.
is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.
b) 7.31
b) 3.57
b) 
b) 
for any integer, a. We can also change any integer to a decimal by adding a decimal point and a zero.
.
(the Greek letter pi, pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat.
.
. list the a) rational numbers b) irrational numbers.
.
and 0.47 are rational.
has no repeating block of digits and it does not stop.
.
b) 
b) 
, list the a) whole numbers b) integers c) rational numbers d) irrational numbers e) real numbers.
.
, are rational. Rational numbers also include fractions and decimals that repeat or stop, so 
are rational. So the list of rational numbers is 
,
.
b) 
c) 
d) 
e)
b) 
c) 
d) 
e)
on the number line.
. We know the proper fraction 
The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts 
. We plot 
is between 0 and 
. These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See 
.
.
. It is easier to plot them if we convert them to mixed numbers and then plot them as described above: 
.
.
.
is to the right of 
on the number line.
b) 
c) 
d) 
.
b) 
c) 
d) 
.
, a proper fraction, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. We write 0 as 0.0 and 1 and 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line. See 
on the number line.
, so it is located between 0 and 
.
.
> 
using < or >.
> 
> 
> 
.
.
using < or >.
> 
, 830 thousandths is greater than 803 thousandths.
> 
> 
.
> 
.
. See 
> 
.
.
> 
.
.
b) 9.279
b) 
b) 
b) 
b) 
b) 
c) 
d)
e) 
c) -7, 
d) none e) 
give the same result as 
and then 
. Others might start with 
and then 
.
.
.
.
. We got the same result both ways, but which way was easier? Multiplying 
.
.
.
.
.
.
.
but we could not multiply 3x without having a value for x.
.
and 
.
and 
.
. Remember, a number and its opposite add to zero.
the multiplicative inverse of a. The reciprocal of 
.
.
, and then simplify to 
. Thus, the additive inverse of 
c) 
.
c)
b) 
c) 
d) 
b) 
c) 
d) 
.
. Therefore, the multiplicative inverse of 9 is 
. Then we find the reciprocal of the fraction. The reciprocal of 
. So the multiplicative inverse of 0.9 is 
c) 
b) 
b) 
c) 
Think about a real example: If there are no cookies in the cookie jar and 3 people are to share them, how many cookies does each person get? There are no cookies to share, so each person gets 0 cookies. So,
.
.
and 
.
means 
. Is there a number that multiplied by 0 gives 4? Since any real number multiplied by 0 gives 0, there is no real number that can be multiplied by 0 to obtain 4
and so we say that division by 0 is undefined.
and 
are undefined.
b) 
c) 
.
b) 
c) 
.
b) 
c) 
.
, where 
b) 
, where 
.
, where 
b) 
, where 
.
b) 
.
.
.
.
.
.
.
.
.
.
, the order of operations says to work in the parentheses first. But we cannot add x and 4, since they are not like terms. So we use the distributive property, as shown in 
.
.
.
.
.
.
.
.
.
.
.
.
. Do you know why?
.
.
.
.
.
.
.
.
.
.
.
are real numbers, then
. A number and its reciprocal multiply to one. 
– The product of any real number and 0 is 0.
– Zero divided by any real number except zero is zero.
c) 0.13
b) 
c) 
c) 
, where 
, where 
, where 
, where 
, where 
, where 
where 
where 
to find the total number of minutes and then multiplying the answer by 
to convert minutes into hours.
.
, and (b), where you multiplied 
.
.
.
.
using the distributive property and explain each step.
and then using the distributive property.
c) 8 d) 
b) 
b) 
c) 
÷
÷
÷
÷
÷
÷
÷
when
when
when 
and 
when 
and 
when
when
___ 
___ 
b) 
answers may vary
answers may vary
b) 0
b) 
b) 
,
d) 
e) 
b) 
c) 14 d) 
b) 
c)
, and 
on a number line.
÷
. When we multiply by this fraction we do not change the value, but just change the units.
also equals 1. How do we decide whether to multiply by 
inches to feet, which multiplication will eliminate the inches?
.
, 
, and 
.
of a metre, just like one cent is 
(or 0.001). This is the same as moving the decimal three places to the left.
°F what does that mean?
.
.
hours for his appointment. Convert the time to seconds.
hours. Convert the time to seconds.
yards
tons
ounces
gallons
little cubes, with each one measuring one inch on all sides. So each little cube has a volume of 
, and the adjacent side as the width, 
. See 
, of the rectangle is the distance around the rectangle. If you started at one corner and walked around the rectangle, you would walk 
units, or two lengths and two widths. The perimeter then is
, we see that the area, 
, is the length, 
.
° angles.
yards and the width is 
feet and the width is 
feet. Find a) the perimeter and b) the area.
inches and width of 
inches.
yards and width of 
inches. Find the length and width.
, this works!
square feet. The length is 
square feet. The length is 
square metres. What is the length?
feet. The length is 
feet. The length is 
feet more than the width. Find the length and width.
yards. Find the length and width.
, so it’s area is 
.
. This example helps us see why the formula for the area of a triangle is 
.
° angle with the base. 
, the sum of the measures of the angles is 
°.
°
inches and whose height is 
feet. The lengths of two sides are 
feet. How long is the third side?
square inches. The base is 
inches. Find the length of each side.
centimetres.
inches of beading to use as trim around a scarf. The scarf will be an isosceles triangle with a base of 
. The height, 
.
square inches.
and 
feet.
and 
centimetres. What is the area?
and 
metres. What is the area?
yards and the bases are 
and 
yards. How many square yards will be available to plant?
yards and 
°
feet and the width is 
feet.
inches and the width is 
inches and width 
yards and width of 
metres and length 
metres and length 
metres.
square metres. The length is 
square centimetres. The width is 
metres less than the length. The perimeter of a rectangle is 
metres. Find the dimensions of the rectangle.
metres less than the width. The perimeter of a rectangle is 
metres. Find the dimensions of the rectangle.
feet. Find the length and width of the rectangle.
metres. Find the length and width.
inches. Find the length and width.
square inches. What is the length?
inches. The area is 
square inches. What is the width?
square metres. The length is 
metres. What is the width?
square feet. The width is 
feet. The length is 
centimetres. The length is 
metres and height 
metres.
feet and height 
feet.
feet. What is its area?
centimetres, how long is the third side?
square inches and height of 
square inches and base of 
centimetres, find the length of each of the other sides.
inches, find the length of each of the other sides.
metres.
inches. The length of the shortest side is 
inches, find the length of each of the other sides.
metres. If the perimeter is 
and 
metres.
and 
centimetres. What is the area?
and 
feet. What is the area?
metres and bases of 
and 
metres.
centimetres and bases of 
and 
centimetres.
inches and lengths 
inches and 
inches. What is the area of the scarf?
per foot. How much will it cost for Christa to fence in her flowerbed?
. About how much will it cost for Caleb to paint the attic wall?
, of any rectangular solid is the product of the length, width, and height.
for 
in the volume formula to get another form of the volume formula.
rectangular solid we started with. See 
of the rectangular solid shown in 
square centimetres.
and 
.
,
inches on each side. Find its a) volume and b) surface area.
.
, can also be used to find the volume of a cylinder.
. 
. We can think of a cone as part of a cylinder. 
, for 
yards, width 
yards, height 
yards
centimetres, width 
centimetres, height 
cm, width 
feet, width 
feet, and height 
feet
metres long. Find its a) volume and b) surface area.
feet
centimetres. Find its a) volume and b) surface area.
inches. Find its a) volume and b) surface area.
yards, height 
centimetres and radius 
metres and radius 
feet across at the base?
feet. To the nearest tenth,
is the length of the diametre
depending on whether the radius of the circle is given as a decimal or a fraction. If you use the 
.
centimetres.
✓
centimetres.
feet.
.
square units.
metres and width 
metres. Find the area enclosed by the track. Round your answer to the nearest hundredth.
feet. What is the circumference of the grill?
inches. What is the circumference of the top?
inches. Find the diametre.
feet. Find the diametre.
metres. Find the diametre.
centimetres. Find the diametre.
feet.
centimetres.
miles.
inches.
feet long, and the triangles are isosceles right triangles. Find the area of the park.
by 
rectangle with a semicircle of diametre 
by 
rectangle with a semicircle at each end, as shown. a) Find the area of the table with one leaf up. b) Find the area of the table with both leaves up.
by 
rectangle, and the ceiling is 
door on one wall, a 
by 
window on the third wall. The fourth wall has no doors or windows. If she will only paint the four walls, and not the ceiling or doors, how many square feet will she need to paint?
feet. Convert the height to miles.
hours. Convert the time to minutes.
minutes. How much time, in hours and minutes, did Pedro spend reading?
metres tall. Convert the height to kilometres.
metres tall. Convert her height to centimetres.
milligrams of calcium. Convert this to grams.
millilitre s. Convert this to litres.
bottle of water. If she drank 
millilitres, how much water, in millilitres, was left in the bottle?
centimetres tall. How much taller, in metres, is Minh than his daughter?
inches tall. Convert his height to centimetres.
kilograms. Convert his weight to pounds.
litres of gas. Convert this to gallons.
centimetres and width of 
square metres. The length is 
square centimetres. What is the length?
centimetres. Find the length and the width.
centimetres and height 
feet. One side of the triangle is 
metres.
and 
centimetres. What is the area of the flag?
inches
inches and height 
°C
°F
milligrams of calcium. Convert this to grams. 
miles. Convert this distance to feet. 
minutes. How much time, in hours and minutes, did Larry spend on the phone? 
to convert 
inches tall. Convert her height to centimetres. Round to the nearest centimetre. 
instead of simplifying it to 
b)
.
b) 
.
b) 
.
. We can write it as a fraction with decimals and then multiply the numerator and denominator by 
and 
. The least common denominator of 
and 
is 
by 
is equivalent to 
b) 
.
b) 
.
as a fraction.
.
.
mg/dl and his HDL cholesterol is 
, so 
. Hector’s cholesterol ratio is high! Hector should either lower his total cholesterol or raise his HDL cholesterol.
mg/dL and HDL cholesterol is 
mg/dL and HDL cholesterol is 
dollars per 
miles in 
miles in 
miles in 
miles. This is correct, but not very useful. We usually want the rate to reflect the number of miles in one hour. A rate that has a denominator of 
for each hour you work, you could write that your hourly (unit) pay rate is 
(read 
last week for working 
. What is Anita’s hourly pay rate?
per hour.
for 
for 
miles, using 
miles to 
miles to 
gallons of gas.
for a case of 
per bottle. Each bottle costs about 
of juice boxes for 
for 
for 
per load and the unit price of the powder detergent is about 
per load. The powder is the better buy.
for 
for 
for 
ounces
miles per 
students to 
dollars for 
miles per 
.
to 
to 
to 
to 
to 
to 
to 
to 
to 
to 
ounces
feet
milligrams to 
milligrams to 
to HDL cholesterol of 
calories per 
miles in 
for 
for 
miles on 
miles on 
magazines in 
sections in 
for 
for 
size for 
size for 
or 
size for 
ounces for 
or 
ounces for 
regular bottle for 
or 
squeeze bottle for 
regular bottle for 
or 
squeeze bottle for 
for 
for a 
for 
miles per 
for 
minutes for 
and 
students and 
pounds of paper trash per year 
How many pounds of paper trash does the average Canadian produce each day? (Round to the nearest tenth of a pound.)
calories, 
grams of fat, 
chocolate mocha coffee with whipped cream contains 
calories, 
or 
Explain your reasoning.
for every 
cup of frozen yogurt and then went for a swim. The frozen yogurt had 
calories. Swimming burns 
calories per hour. For how many minutes should Kathryn swim to burn off the calories in the frozen yogurt? Explain your reasoning.
calories. If Arjun walks for one hour, he burns 
to show percent.
of Canadian Internet users reported a cyber security incident, including being redirected to fraudulent websites that asked for personal information or getting a virus or other computer infection. This means 
.
means a ratio of 
means a ratio of 
and 
means a ratio of 
. In words, “one hundred percent” means the total 
, we see that 
of Canada’s Universities are working to include Indigenous representation within their governance or leadership structures.Write this percent as a ratio.
of college students have a smartphone.
of Canadian teens send text messages regularly.
, according to a Universities Canada survey, 
means 
.
sign. (Sometimes the decimal point does not appear in the percent number, but just like we can think of the integer 
, we can think of 
.) Notice that we may need to add zeros in front of the number when moving the decimal to the left.
believe that poor math and science education in Canada will lead to higher unemployment rates.
when one celebrity tweeted live on air.
of adults aged 
.
b) 
b)
.
b)
b) 
> 
b)
b)
b) 
c) 
b) 
c) 
b)
c)
. We will use this same notation, as well as fraction notation, when we convert fractions to percents in the next example.
.
people within the population of Canada are between the ages of 
, the unemployment rate for those with only a high school degree was 
, among the unemployed, 
were long-term unemployed.
in 
was 
.
of home sales had owner financing. (Source: Bloomberg Businessweek, 5/23–29/2011)
of the Canadian population was a visible minority. (Source: www12.statcan.gc.ca)
the number of cell phones in India was 
of the population.
are women.
.
.
of Canadian adults were overweight or obese.
of Canadian workers are using more than one language at work.
. She knows this is a little less than 
dress.
paycheck.
rating as a fraction and a decimal.
rating as a percent and a decimal.
rating as a percent and a decimal.
to fractions. Do you notice a pattern? Explain what the pattern is.
, and 
of what they had paid for the house 
ago. Explain what 
was 
of what it had been in 
. Explain what 
.
, where 
.
is to 
is a proportion because the two fractions are equal. The proportion 
is to 
we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.
is to 
hits in 
for 
for 
hits in 
at-bats.
for 
for 
is to 
adults for 
for 
for 
. From our work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers?
, the equation is not a proportion.
, the equation is a proportion.
.
.
.
.
. Try it and verify that you get the same solution.
.
.
.
.
.
ml of acetaminophen.
number of calories.
calories. How many calories will she get if she drinks the large 
dollars into Mexican pesos. At that time, the exchange rate had 
U.S. is equal to 
Mexican pesos. How many Mexican pesos did he get for his trip?
number of pesos.
dollars into Euros. At the current exchange rate, 
into Japanese yen. If each dollar is 
yen, how many yen will they get?
and we can simplify 
. Since the equation 
of 
.
of what number?
.
of 
number.
of 
of 
of what number is 
of what number is 
of what number is 
.The proportion states two ratios or rates are equal. The proportion is read “
.
.
games.
for 
for 
for 
for 
for 
for 
for 
for 
calories for 
Canadian dollars into US dollars. At the current exchange rate, 
US. How many Canadian dollars will she get for her trip?
into Mexican pesos. If each dollar is worth 
pesos, how many pesos will he get for his trip?
into 
Euros. How many Euros did he receive per Canadian dollar?
Canadian into 
Australian dollars. How many Australian dollars did she receive per US dollar?
into quarters. How many quarters did she get?
into nickels. How many nickels did she get?
miles away, how many gallons of gas are needed to get there and then home? If gas is 
per gallon, what is the total cost of the gas for the trip?
miles from Danny’s home and his car gets 
miles away. After 
miles. At that rate, how long will the whole drive take?
miles. After 
miles. At that rate, how long will the whole drive take?
square feet of lawn. Phil’s lawn is approximately 
square feet. How many bags of fertilizer will he have to buy?
square feet. How many gallons of paint will she have to buy?
of 
of 
of what number?
is 
is 
of 
of 
of 
is what number?
of 
is 
of what number?
is 
of what number?
do you prefer to use an equation like you did in the section on Decimal Operations or a proportion like you did in this section? Explain your reason.
is 
do you prefer to use an equation like you did in the section on Decimal Operations or a proportion like you did in this section? Explain your reason.
. They wanted to leave a 
tip. What amount would the tip be?
, or 
the number.
is 
of 
the number.
the number.
of what number?
is what percent of 
. They want to leave an 
amount of tip.
. If she wants to leave 
of the total bill as her tip. If her bill was 
, how much will she leave for the tip?
calories, and had 
calories and 
, university tuition fees in Canada for domestic students increased from 
per school year to 
per school year. Find the percent increase. (Round to the nearest tenth of a percent.)
, the IRS increased the deductible mileage cost to 
cents from 
, the standard bus fare in Vancouver was 
. In 
. The average price in that city in July was 
. Find the percent decrease.
in 
. The population of the city is projected to be about 
in 
. Find the percent decrease. (Round to the nearest tenth of a percent.)
. Because of furlough days, this year her salary was 
. Find the percent decrease. (Round to the nearest tenth of a percent.)
is what number?
of 
is what number?
is what number?
of what number is 
of what number is 
of what number is 
is 
is 
is what percent of 
is what percent of 
. She wants to leave 
of the total bill as a tip. How much should the tip be?
. They want to leave 
of each paycheck automatically deposited to his savings account. His last paycheck was 
. How much money was deposited to Trong’s savings account?
. How much did Cherise deposit into her retirement account?
of the recommended daily amount. What is the total recommended daily amount of fiber?
milligrams (mg) of sodium, which is 
of the recommended daily amount. What is the total recommended daily amount of sodium?
of the recommended daily amount. What is the total recommended daily amount of sodium?
calories, and 
calories are from fat. What percent of the total calories is from fat?
calories are from fat. What percent of the total calories is from fat?
per month. She pays 
a month for rent. What percent of her monthly pay goes to rent?
per month. She pays 
a month for rent. What percent of her monthly pay goes to rent?
to 
. Find the percent increase.
to 
. Find the percent increase.
in 
in 
. Find the percent increase.
annual global seafood consumption rose from 
to 
to 
. Find the percent increase. (Round to the nearest tenth of a percent.)
to 
. Find the percent decrease.
to 
. Find the percent decrease.
last year. This year his salary was cut to 
. Find the percent decrease.
to about 
. Find the percent decrease. (Round to the nearest tenth of a percent.)
to 
. Find the percent decrease. (Round to the nearest tenth of a percent.)
million to 
million in one year. Find the percent decrease. (Round to the nearest tenth of a percent.)
. MaryAnne brings 
with her when she buys a cup of coffee and leaves the change as a tip. What percent tip does she leave?
. She was charged a 
is 
Explain your reasoning.
think about what the solution might be. Should it be a number that is greater than 
more shirts. Explain what Alex meant.
for 
. How much did each towel cost? Round to the nearest cent if necessary.
. The box contains 
or 
miles per 
and the difference of 
admission rate for the university
rate of college students with student loans
of the Canadian population was age 
of the Canadian population was under 
of 
of 
of what number is 
is 
. Aurelio wants to leave 
per month. Her car payment is 
. What percent of her monthly pay goes to her car payment?
from her uncle. She spent 
. If the value of the car depreciated 
miles to 
for 
for 
into Euros. If each dollar is 
miles. If his car gets 
of 
of what number?
is 
is 
of what number?
is 
miles.
for 
. How much did each brush cost? Round to the nearest cent if necessary
is what percent of 
into Polish zlotych. If each dollar is 
zlotych, how many zlotych will Klaudia get?
is a solution for 
.
a solution for 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
into the original equation.
.
.
.
.
.
.
and 
is 
and 
and 
is 
✓
for her new car. This was 
less than the sticker price. What was the sticker price of the car?
the sticker price of the car.
✓
for his new car. This was 
less than the sticker price. What was the sticker price of the car?
. This is 
less than the price at night. How much does the movie cost at night?
a solution of 
a solution of 
a solution of 
and 
is 
and 
is 
and 
is 
degrees. What was his temperature last night?
pounds. How much did Tricia’s daughter weigh at her last checkup?
less than his paycheck last week. His paycheck this week was 
. How much was Ron’s paycheck last week?
less than her art book cost. Her math book cost 
. How much did her art book cost?
screw. The screw should be 
to see what size the hole should be.
to find the amount of sugar she should ask to borrow.
and then make up an application that uses this equation in its solution.
How do you know?
or 
. We were able to isolate the variable by adding or subtracting the constant term on the side of the equation with the variable. Now we will see how to solve equations that have a variable multiplied by a constant and so will require division to isolate the variable.
).
, so it works! Three counters in each of two envelopes does equal six!
.
.
.
.
.
.
.
.
.
.
.
is equivalent to 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
for a pack of 
.
for 
.
.
. Find the miles per gallons, 
.
is the solution to the equation 
. Explain why he is wrong.
by adding 
. We already know how to solve equations of this form by using the Division or Multiplication Properties of Equality.
.
.
.
.
.
, so to ‘undo’ subtraction, add 
.
.
.
.
.
.
.
from both sides.
.
.
.
.
.
.
.
and 
. Since 
.
.
.
on the left and 
on the right. Since 
to both sides to leave only constants on the right.
.
.
.
on the left and 
on the right. Since 
.
.
.
.
from both sides.
.
.
.
, make the left side the variable side and the right side the constant side.
from both sides.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Student tickets sold for 
and adult tickets sold for 
The number of adult tickets sold was 
.
in nickels and dimes. The number of dimes is one less than twice the number of nickels. Find the number of nickels, 
.
.
, explaining all the steps of your solution.
Explain why this is your first step.
explaining all the steps of your solution as in the examples in this section.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
and 
. So, when we have an equation with decimals, we can use the same process we used to clear fractions—multiply both sides of the equation by the 
.
.
.
.
.
.
.
.
.
.
.
for 
worth of 
stamps and 
stamps. The number of 
for 
.
, what is the LCD? How do you know?
.
.
as 
.
.
.
.
.
.
.
.
![10\left[3-8\left(2s-5\right)\right]=15\left(40-5s\right)](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-17da43c7e64c31b1dc3e32d5bf001e61_l3.svg "Rendered by QuickLaTeX.com")
.
to get the s’s to the right.
.
![6\left[4-2\left(7y-1\right)\right]=8\left(13-8y\right)](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-1672f14cc5f2e201d309ef003cc66c9d_l3.svg "Rendered by QuickLaTeX.com")
.
![12\left[1-5\left(4z-1\right)\right]=3\left(24+11z\right)](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-5a8ef9f112dca68f37de134b453a290c_l3.svg "Rendered by QuickLaTeX.com")
.
.
to get the variables to the left.
.
. The solution we found was 
for a different value of x, the left side will not be 
. Do you recognize that the left side and the right side are equivalent? Let’s see what happens when we solve for y.
is true.
?
to get the constant alone on the right.
’s are gone!
.
. The equation 
to get the 
.
from both sides.
.
![5\left[9-2\left(6d-1\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-7a485a000b683ec41becb4471d58479d_l3.svg "Rendered by QuickLaTeX.com")
![4\left[5-8\left(4c-3\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-d31b0be6e5230f88e857248e4a3fd862_l3.svg "Rendered by QuickLaTeX.com")
![3\left[-14+2\left(15k-6\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-96e8ff1b9ee38cf5d5dcccbb7fa6911d_l3.svg "Rendered by QuickLaTeX.com")
![3\left[-9+8\left(4h-3\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-dd07d0cc8d1c746b864c9bc256158349_l3.svg "Rendered by QuickLaTeX.com")
![10\left[5\left(n+1\right)+4\left(n-1\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-b19b83d20ec2f3b76a9b1166884a2e20_l3.svg "Rendered by QuickLaTeX.com")
![=11\left[7\left(5+n\right)-\left(25-3n\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-c29f200d533fe721e432ffcb6dff9005_l3.svg "Rendered by QuickLaTeX.com")
![5\left[2\left(m+4\right)+8\left(m-7\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-7f44ec79ad6af354e021a613de8f3ac9_l3.svg "Rendered by QuickLaTeX.com")
![=2\left[3\left(5+m\right)-\left(21-3m\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-ccaf5bd423fe1b36358d0ea6cb9f4f9a_l3.svg "Rendered by QuickLaTeX.com")
.
? Why is this your first step?
miles
hours at 60 miles per hour. How much distance did she travel?
hours at 3 miles per hour. How far did she walk?
, the rate and time. But in 
and then used algebra to solve for 
and 
is solved for 
b) in general
b) 
b) in general
and 
b) in general
.
and 
b) in general
b) 
and 
b) in general
b) 
is used to calculate simple interest, I, for a principal, P, invested at rate, r, for t years.
, 
, 
, 
, 
b) in general
, 
, 
b) in general
for y:
for y:
b) in general
for y:
b)
for y.
by 5
for y.
for y.
hours at 60 miles per hour. How much distance did she travel?
hours at 72 miles per hour. How much distance did he travel?
hours at 12 miles per hour. How far did she ride?
and 
,
,
,
,
for y
,
,
for y
for y
for 
for y
for 
for y.
for y.
for y.
for y.
for y.
for y.
for y.
for y.
.
to find the Fahrenheit temperature.
for 
for 
b) 
b) 
b) 
b) 
b) 
b) 
years b) 
b) 
b) 
, which means “the original price” was $36
✓
number.
number
✓
✓
number
✓
✓
number.
number
✓
✓
, which is 
next consecutive integer
3rd consecutive integer
.
or 
.
even integer.
consecutive even integer
consecutive even integer
the amount the husband earns.
the amount the wife earns.
3h − 16,000 = 110,000
3h = 126,000
h = 42,000
$42,000 amount husband earns
2(42,000) − 16,000
84,000 − 16,000
68,000
. One number is 35 more than the other. Find the numbers.
. One number is 94 less than the other. Find the numbers.
. One number is 62 less than the other. Find the numbers.
.
.
![2\left[-16+5\left(8k-6\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-f3c715a499010cfd8f8b08dd2c01073a_l3.svg "Rendered by QuickLaTeX.com")
hours at 60 miles per hour. How much distance did she travel?
and 
and 
and 
and 
for y
,
,
; b)
b)
b)
.
![9\left(3u-2\right)-4\left[6-8\left(u-1\right)\right]](https://opentextbc.ca/introalgebra/wp-content/ql-cache/quicklatex.com-13c5398ad12d06eee66411cefef634da_l3.svg "Rendered by QuickLaTeX.com")
for y
b) 
Graphs are found in all areas of our lives—from commercials showing you which cell phone carrier provides the best coverage, to bank statements and news articles, to the boardroom of major corporations. In this chapter, we will study the rectangular coordinate system, which is the basis for most consumer graphs. We will look at linear graphs, slopes of lines, and equations of lines.
, gives the coordinates of a point in a rectangular coordinate system.
. The point 
as an example. First, locate 1 on the x-axis and lightly sketch a vertical line through 
B 
C 
D 
E 
.
, the point is to the right of the y-axis. Since 
, the point is above the x-axis. (It may be helpful to write 
B 
C 
D 
E 
.
B 
D 
E 
located? Where is the point 
located?
.
.
B 
C 
D 
.
, the point whose coordinates are 
E 
.
B 
C 
.
.
.
. The coordinates of point E are 
.
B: 
C: 
D: 
E: 
F: 
B: 
D: 
E: 
. Equations of this form are called linear equations in two variables.
is also a 
be integers and 
when writing a linear equation in standard form, although it is not strictly necessary.
.
C 
C 
? A 
B 
?
C 
? A 
B 
C 
? A 
B 
is a solution to 
.
.
.
.
.
.
.
and 
are all solutions to 
.
.
.
.
in a rectangular coordinate system.
D: 
is on the line shown in 
is not a solution to the equation 
is shown.
C 
by plotting points.
.
.
.
.
.
.
.
, which we found by letting 
.
.
.
.
.
.
.
.
. What is the difference between the equations 
in the same rectangular coordinate system.
and 
and 
and 
gives the cost, 
gives the amount, 
.
. The y– intercept is 
.
, y– intercept: 
.
.
.
using the intercepts.
using the intercepts.
using the intercepts.
using the intercepts.
, find the intercepts. Find the x– and y– intercepts and then a third point.
? Why?
? Why?
, he has been gone 0 hours and has 1000 miles left. At 
, he has been gone 15 hours and has 0 miles left to go.
to find the slope of a line from its graph
.
. Draw a picture to show your results.
. Draw a picture to show your results.
.
. Draw a picture to show your results.
. Draw a picture to show your results.
.
and 
, and that simplifies to 
.
.
to identify the first point and 
to identify the second point.
, 
, and so on.
.
.
, the y-coordinate of the second point and 3 is 
, the y-coordinate of the first point.
, the x-coordinate of the second point and 2 is 
, the x-coordinate of the first point.
.
.
.
.
.
.
.
.
.
.
and 
.
.
.
. Go back to 
.
.
.
. Go back to 
.
.
; 
; 
, where 
; rise = 3, run = 50
. Compare these values to the equation 
.
.
)
using its slope and y-intercept.
using its slope and y-intercept.
using its slope and y-intercept.
using its slope and y-intercept.
using its slope and y-intercept.
using its slope and y-intercept.
using its slope and y-intercept.
using its slope and y-intercept.
using its slope and y-intercept.
. Given the scale of our graph, it would be easier to use the equivalent fraction 
.
using its slope and y-intercept.
using its slope and y-intercept.
c) 
d) 
d) 
c) 
.
, on the Fahrenheit scale.
.
.
, means that the temperature Fahrenheit (F) increases 9 degrees when the temperature Celsius (C) increases 5 degrees.
then count out the rise of 9 and the run of 5 to get a second point. See 
is used to estimate a woman’s height in inches, h, based on her shoe size, s.
is used to estimate the temperature in degrees Fahrenheit, T, based on the number of cricket chirps, n, in one minute.
models the relation between her weekly cost, C, in dollars and the number of pizzas, p, that she sells.
.
.
models the relation between his weekly cost, C, in dollars and the number of miles, m, that he drives.
models the relation between her weekly cost, C, in dollars and the number of wedding invitations, n, that she writes.
, and different y-intercepts.
and 
are the slopes of two parallel lines then
.
on the same grid. The first equation is already in slope–intercept form: 
and 
are parallel.
and 
are parallel.
and 
are parallel.
.
and 
are parallel. You may want to graph these lines, too, to see what they look like.
and 
are parallel.
and 
and 
, shown in 
, and the slope of the second line, 
, we can see that they are negative reciprocals of each other. If we multiply them, their product is 
and 
and 
are perpendicular.
are perpendicular.
are perpendicular.
and 
are perpendicular.
and 
are perpendicular.
and 
are perpendicular.
are the slopes of two perpendicular lines, then 
and y-intercept, 
is, 
.
models the relation between the amount of Randy’s monthly water bill payment, P, in dollars, and the number of units of water, w, used.
models the relation between the amount of Tuyet’s monthly water bill payment, P, in dollars, and the number of units of water, w, used.
models the relation between the cost in dollars, C, per day and the number of miles, m, she drives in one day.
models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day.
models the relation between his weekly salary, S, in dollars and the amount of his sales, c, in dollars.
models the relation between her weekly salary, S, in dollars and the amount of her sales, c, in dollars.
models the relation between the cost in dollars, C of the banquet and the number of guests, g.
models the relation between the cost in dollars, C, of the banquet and the number of guests, g.
is used to estimate the number of cricket chirps, n, in one minute based on the temperature in degrees Fahrenheit, T.
can be used to convert temperatures F, on the Fahrenheit scale to temperatures, C, on the Celsius scale.
chirps when the Fahrenheit temperature is 
.
. Write the equation in slope–intercept form.
and containing the point 
.
.
. Write the equation in slope–intercept form.
.
.
.
?
.
.
and 
. Write the equation in slope–intercept form.
.
.
. We’ll use the notation 
to represent the slope of a line parallel to a line with slope 
looks like two parallel lines.)
), we count out the rise 2 and the run 1. We draw the line.
that contains the point 
to represent the slope of a line perpendicular to a line with slope 
.
.
. Write the equation in slope–intercept form.
. Write the equation in slope–intercept form.
.
and y-intercept 
and y-intercept 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
and 
and 
and 
and 
and 
and 
and 
and 
and 
and 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, point 
, containing point 
and
. Find a linear equation that models the relationship between city mpg and highway mpg.
and 
. Find a linear equation that models the relationship between age and LDL cholesterol level.
and 
to find the slope of a line from its graph.
models the relation between her weekly profit, P, in dollars and the number of student lessons, s, that she teaches.
models the relation between her weekly cost, C, in dollars and the number of meals, m, that she serves.
, point 
and 
and 
and 
, point 
, point 
, point 
, point 
d) 
and y-intercept 
?
.
and 
, containing the point 
means to multiply 2 by itself 4 times, so 
power.
, the exponent 
b) 
c) 
d) 
.
b) 
c) 
d) 
.
d) 0.1849
b) 
c) 
d) 
.
d) 
b) 
.
.
b) 
.
b) 
to the 4th power. In part b) we raise just the 5 to the 4th power and then take the opposite.
.
.
.
b) 
.
b) 
.
b) 
b) 
.
b) 
b) 
.
b) 
.
b) 
b) 
.
b) 
.
.
.
b) 
.
b) 
.
b) 
b) 
.
b) 
b) 
.
b) 
.
b) 
b) 
.
b) 
b) 
.
b) 
.
b) 
b) 
.
b) 
b) 
.
to the second power.
b) 
.
b) 
b) 
.
b) 
, where 
, and 
.
.
.
.
.
.
.
are whole numbers, then
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
, on the third round is 
, as shown in the table below. How many people will receive the email on the sixth round? Simplify the expression to show the number of people who receive the email.
, after 2 years was 
, after 3 years was 
, as shown in the table below. What will Jamal’s salary be after 10 years? Simplify the expression, to show Jamal’s salary in dollars.
and the cost of the item during the second week would be 
. Complete the table shown below. What will be the cost of the sofa during the fifth week? Simplify the expression, to show the cost in dollars.
and the value of the car after the end of the second year would be 
. Complete the table shown below. What will be the value of the car at the end of the eighth year? Simplify the expression, to show the value in dollars.
.
but 
.
is 1. What is wrong with his reasoning?
is 
, and not 
d) 0.125
d) 0.064
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
and 
> 
> 
b) 
.
.
b) 
.
b) 
.
b) 
b) 
.
.
b) 
.
b) 
b) 
.
b) 
b) 
.
in the denominator and so we will end up with factors in the denominator.
b) 
.
b) 
b) 
.
b) 
. From your earlier work with fractions, you know that:
, for any 
, since any number divided by itself is 1
when 
?
and to 1. So 
.
b) 
.
b) 
.
b) 
.
. We can use the product to a power rule to rewrite this expression.
b) 
.
b) 
.
b) 
.
c) 
.
b) 
c) 
.
b) 
c) 
b) 
c) 
.
b) 
c) 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
by dividing out a common factor, and to simplify the variables by subtracting their exponents.
.
.
.
.
.
> 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
c) 
b) 
c) 
b) 
b) 
bytes. One gigabyte is approximately 
bytes. How many megabytes are in one gigabyte?
bytes. How many gigabytes are in one terabyte?
simplifies to 
. What is wrong with her reasoning?
by writing 
. What is wrong with his reasoning?
and 
he got the same answer. Explain how using the Order of Operations correctly gives different answers.
simplifies to 0. What would you say to convince Robert he is wrong?
b) 
b) 
b) 
b) 
c) 
b) 
are whole numbers, then
.
and it leads us to the definition of a negative exponent.
.
, we will take one more step and write 
. The answer is considered to be in simplest form when it has only positive exponents.
b) 
.
b) 
.
b) 
.
.
b) 
.
b) 
.
b) 
b) 
.
b) 
.
, and 
.
b) 
.
b) 
.
b) 
b) 
.
b) 
b) 
c) 
d) 
.
.
.” Here the exponent applies to the base 
.
b) 
c) 
d) 
.
c) 25 d) 
b) 
, c) 
d) 
.
b) 
c) 49 d) 
b) 
.
b) 
.
b) 
.
b) 
.
b) 
.
b) 
b) 
.
b) 
b) 
c) 
.
.
b) 
c) 
.
b) 
c) 
b) 
c) 
.
b) 
c) 
, for 
b) 
c) 
.
.
b) 
c) 
.
c) 
b) 
c) 
.
b) 
c) 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. We know that 4,000 means 
and 0.004 means 
.
multiplication sign, even though we avoid using this sign elsewhere in algebra.
.
.
.
.
.
.
.
.
.
.
.
. Write answers in decimal form.
. Write answers in decimal form.
.
. Write answers in decimal form.
. Write answers in decimal form.
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b)
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
b) 
. Convert this number to decimal form.
. Convert this number to decimal form.
. Convert this number to decimal form.
of the width of an atom. Convert this number to decimal form.
b) 
b) 
b) 25
b) 10000
b) 
b) 
b) 
c) 
b) 
b) 
b) 
b) 
c)
b) 
c) 
b) 
c) 
.
b) 
c) 
For example, 
is read as 
. Similarly, 
is the square of 
.
, then 
, we say 
also, so 
, stands for the positive square root. The positive square root is also called the principal square root.
is read as “the square root of 
.
. Notice that zero has only one square root.
b) 
.
b) 
.
b) 
.
. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, 
.
b) 
b) 
.
b) 
.
Is there a number whose square is 
. If we are asked to find the 
b) 
.
is not a real number.
b) 
.
b) 
b) 
.
b) 
.
b) 
.
is between 
between two consecutive whole numbers.
between two consecutive whole numbers.
between two consecutive whole numbers.
key on your calculator. You will to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator’s display. The symbol for an approximation is 
and it is read approximately.
display. Using it to find the square root of 
. This is the approximate square root of 
. The square root of 
to two decimal places using a calculator.
to two decimal places.
to two decimal places.
, where 
. Can you think of an expression whose square is 
.
.
.
.
.
.
.
.
.
.
.
.
units. See the table below.
.
.
feet.
square centimetres. How long can a side of his mosaic be?
. For example, if an object is dropped from a height of 
.
.
feet. How many seconds does it take for the package to reach the ground?
feet above the sidewalk. How many seconds does it take for the squeegee to reach the sidewalk?
.
when
.
.
feet long. To the nearest tenth, how fast had the vehicle been going before the brakes were applied?
, then 
, for 
. 
square feet. How long can a side of her patio be?
feet above a lake. How many seconds did it take for the flare to reach the water?
feet. What was the speed of the car before the brakes were applied?
feet. What was the speed of the vehicle before the brakes were applied?
tiles. Each tile is square with an area of one square inch. How long can a side of the mosaic be?
and 
reads: “the square root of nine” which gives us the number such that if it were multiplied by itself would give you the number inside of the square root.
is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use in 
? If we simplify the square root first, we’ll be able to estimate it easily. There are other reasons, too, to simplify square roots as you’ll see later in this chapter.
is considered simplified if 
is simplified. But 
is not simplified, because 16 is a perfect square factor of 32
.
.
.
, which is the product of an integer and a square root. We always write the integer in front of the square root.
.
.
.
.
.
.
.
.
.
.
.
.
.
cannot be simplified—to begin we’d need to simplify each square root, but neither 17 nor 7 contains a perfect square factor.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
to determine the length and width of the new garden. Round to the nearest tenth of a foot.
to determine the number of seconds it takes for the sunglasses to reach the ground.
to determine the number of seconds it takes for the sunglasses to reach the ground.
. Then explain why 
.
is not equal to 
.
70)
b)
÷ 
(6.2 × 
(4 ×
square inch tiles. He wants to use them to make a square mosaic. How long can each side of the mosaic be? 
to decimal form.
b) 
c) 
d) 
e)
b) 
c) 
e) 
b) 
c) 
d) 
e)
c) 
d) 
e)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
from 
.
from 
.
from 
.
.
.
.
.
.
.
.
.
.
when
when
when
gives the height of a ball 
seconds.
seconds.
gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with 
.
.
from 
.
from 
.
from 
.
from 
.
and 
.
and 
.
and 
.
and 
.
and
.
and
.
and
.
and
.
when:
when:
when:
when:
gives the height of the brush 
. Find the revenue received when 
dollars.
dollars.
. Find the fuel efficiency when 
.
. Find the stopping distance when 
.
. Find the cost to rent the cleaner for 6 days.
where 
seconds.
where 
°.
is 
. What is wrong with her reasoning?
. You multiplied both terms in the parentheses, 
. With this chapter’s new vocabulary, you can say you were multiplying a binomial, 
, by a monomial, 2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
by 
.
.
.
.
.
and 15 as 
.
by the FOIL method.
without using a calculator.
and 17 as 
.
by the FOIL method.
without using a calculator.
.
by itself.
by itself.
= ____ + ____ + ____
+ ____ + ____
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. This is called a difference of squares.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
b) 
c) 
d) 
We are asked to square a binomial. It fits the binomial squares pattern.
b) 
c) 
d) 
b) Binomial Squares; 
c) Binomial Squares; 
d) Product of Conjugates; 
b) 
c) 
d) 
b) Product of Conjugates; 
c) FOIL; 
d) Binomial Squares; 
.
by using the binomial squares pattern, 
and 53 as 
.
by using the product of conjugates pattern, 
without using a calculator.
result in a binomial?
is 64, and then use that numerical example to explain why 
.
b) 
c) 
d) 
b) 
c) 
d) 
and 
.
.
.
.
.
and 
is 
.
.
.
.
, 
and 
is 
.
.
or 
, in algebra, it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial as follows:
, and end with its factors, 
. To do this we apply the Distributive Property “in reverse.”
.
.
.
.
.
.
.
.
.
.
, 
and 28.
.
.
.
and 
.
.
.
.
.
.
, 
, 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
, where 
width. Factor the greatest common factor from the polynomial.
. Factor the greatest common factor from the polynomial.
, and 
, and 
.
, and end with the factors, 
. You need to think about where each of the terms in the trinomial came from.
and 
. We factored it into two binomials of the form 
and 
.
.
.
.
.
.
.
.
.
.
.
.
.
, we need two negative factors. Find two numbers that multiply 28 and add to 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
as the last terms of the binomials.
.
.
. The first term, 
. The 
.
.
.
.
.
.
in the second term. The last term of the trinomial is negative, so the factors must have opposite signs.
.
.
.
describes how these numbers are related. Factor the trinomial.
describes how these numbers are related. Factor the trinomial.
as 
. Bill factored it as 
. Phil factored it as 
. Who is correct? Explain why the other two are wrong.
,
.
and 
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
, where 
.
and explain with words how you get each term of the quotient.
by 6 this way: 
. What is wrong with his reasoning?
when:
when:
. Find the revenue received when 
dollars.
gives the height of a stone 
3
b) 10 c) 22
gives the height of the package 
seconds.
It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is 
is the angle with vertex at point 
.
.
.
are supplementary, then 
°.
°.
°. Find a) its supplement, and b) its complement.
°. Find its: a) supplement b) complement.
°. Find its: a) supplement b) complement.
° more than the smaller angle. Find the measure of both angles.
° more than the smaller angle. Find the measures of both angles.
’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.
and sides 
° and 
°. Find the measure of the third angle.
° and 
°. Find the measure of the third angle.
° and 
°. Find the measure of the third angle.
°. What is the measure of the third angle?
°. What is the measure of the other angle?
°. What is the measure of the other angle?
° more than the measure of the smallest angle. Find the measures of all three angles.
° more than the measure of the smallest angle. Find the measures of all three angles.
and their corresponding angles have equal measures.
corresponds to the side 
, we will use the ratio 
to find the other sides.
BCE.
wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.
ladder 
string of lights to the top of the 
mast of his sailboat. How far from the base of the mast should he attach the end of the light string?
.
.
°
°
°
°
° angle.
° angle.
° angle.
° angle.
° less than the larger angle. Find the measures of both angles.
° less than the larger angle. Find the measures of both angles.
° more than the smaller angle. Find the measures of both angles.
° and 
°. Find the measure of the third angle.
° and 
°. Find the measure of the third angle.
° and 
° and 
°. What is the measure of the other angle?
°. What is the measure of the other angle?
°. What is the measure of the other angle?
miles.
string of lights will be attached to the top of a 
pole for a holiday display. How far from the base of the pole should the end of the string of lights be anchored?
extension ladder to paint his house. If he sets the base of the ladder 
to point 
, as shown in the picture below. To find the length of the footbridge, she draws a right triangle 
, with right angle at 
feet, and from 
feet. How long will the bridge be?
from Example 1 to find the three ratios.
or 
. Using the definition of trigonometric ratios, we can write sinE=
.
=0.6
= 1.3333…
= 0.75
= 0.8621
=0.5172
8.03 
14 
= 70.9° Rounded to one decimal place
= 90°
= 68°
= 59.2°
= 18.2°
. Round your final answer to one decimal place.
, cos θ = 
, tan θ = 
, cos θ = 
, tan θ = 
= 90°
= 53.1°
° angle?
° less than the larger angle. Find the measures of both angles.
degrees. What is the measure of the other small angle?
° more than the measure of the smallest angle. Find the measures of all three angles.
° angle?
is similar to 
. Find the missing sides. 
= 27°