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Study Guides > Prealgebra

Problem Set 3: Integers

Practice Makes Perfect

Locate Positive and Negative Numbers on the Number Line

In the following exercises, locate and label the given points on a number line.

22
2-2
5-5
This figure is a number line. Negative 5 is labeled with c, two units to the left of 0 is labeled b, and two units to the right of 0 is labeled a.
  1. 55
  2. 5-5
  3. 2-2
  1. 8-8
  2. 88
  3. 6-6
This figure is a number line. Negative 8 is labeled a, negative 6 is labeled c, and 5 is labeled b.
  1. 7-7
  2. 77
  3. 1-1

Order Positive and Negative Numbers on the Number Line

In the following exercises, order each of the following pairs of numbers, using <<; or >.\text{>.}

9\text{__}4
-3\text{__}6
-8\text{__}-2
1\text{__}-10
  1. >
  2. <
  3. <
  4. >
  1. 6\text{__}2;
  2. -7\text{__}4;
  3. -9\text{__}-1;
  4. 9\text{__}-3
  1. -5\text{__}1;
  2. -4\text{__}-9;
  3. 6\text{__}10;
  4. 3\text{__}-8
  1. <
  2. >
  3. <
  4. >
  1. -7\text{__}3;
  2. -10\text{__}-5;
  3. 2\text{__}-6;
  4. 8\text{__}9

Find Opposites

In the following exercises, find the opposite of each number.

  1. 22
  2. 6-6
  1. −2
  2. 6
  1. 99
  2. 4-4
  1. 8-8
  2. 11
  1. 8
  2. −1
  1. 2-2
  2. 66

In the following exercises, simplify.

(4)-\left(-4\right)

4

(8)-\left(-8\right)

(15)-\left(-15\right)

15

(11)-\left(-11\right)

In the following exercises, evaluate.

mwhen-m\text{when}

m=3m=3
m=3m=-3
  1. −3
  2. 3

pwhen-p\text{when}

p=6p=6
p=6p=-6

cwhen-c\text{when}

c=12c=12
c=12c=-12
  1. −12;
  2. 12

dwhen-d\text{when}

d=21d=21
d=21d=-21

Simplify Expressions with Absolute Value

In the following exercises, simplify each absolute value expression.

  1. 7|7|
  2. 25|-25|
  3. 0|0|
  1. 7
  2. 25
  3. 0
  1. 5|5|
  2. 20|20|
  3. 19|-19|
  1. 32|-32|
  2. 18|-18|
  3. 16|16|
  1. 32
  2. 18
  3. 16
  1. 41|-41|
  2. 40|-40|
  3. 22|22|

In the following exercises, evaluate each absolute value expression.

  1. xwhenx=28|x|\text{when}x=-28
  2. uwhenu=15|-u|\text{when}u=-15
  1. 28
  2. 15
  1. ywheny=37|y|\text{when}y=-37
  2. zwhenz=24|-z|\text{when}z=-24
  1. pwhenp=19-|p|\text{when}p=19
  2. qwhenq=33-|q|\text{when}q=-33
  1. −19
  2. −33
  1. awhena=60-|a|\text{when}a=60
  2. bwhenb=12-|b|\text{when}b=-12

In the following exercises, fill in <,>,or=\text{<},\text{>},\text{or}= to compare each expression.

  1. -6\text{__}|-6|
  2. -|-3|\text{__}-3
  1. <
  2. =
  1. -8\text{__}|-8|
  2. -|-2|\text{__}-2
  1. |-3|\text{__}-|-3|
  2. 4\text{__}-|-4|
  1. >
  2. >
  1. |-5|\text{__}-|-5|
  2. 9\text{__}-|-9|

In the following exercises, simplify each expression.

84|8 - 4|

4

96|9 - 6|

878|-7|

56

555|-5|

157146|15 - 7|-|14 - 6|

0

178134|17 - 8|-|13 - 4|

182(83)18-|2\left(8 - 3\right)|

8

153(85)15-|3\left(8 - 5\right)|

8(1422)8\left(14 - 2|-2|\right)

80

6(1342)6\left(13 - 4|-2|\right)

Translate Word Phrases into Expressions with Integers

Translate each phrase into an expression with integers. Do not simplify.

  1. the opposite of 88
  2. the opposite of 6-6
  3. negative three
  4. 44 minus negative 33
  1. −8
  2. −(−6), or 6
  3. −3
  4. 4−(−3)
  1. the opposite of 1111
  2. the opposite of 4-4
  3. negative nine
  4. 88 minus negative 22
  1. the opposite of 2020
  2. the opposite of 5-5
  3. negative twelve
  4. 1818 minus negative 77
  1. −20
  2. −(−5), or 5
  3. −12
  4. 18−(−7)
  1. the opposite of 1515
  2. the opposite of 9-9
  3. negative sixty
  4. 1212 minus 55

a temperature of 6degrees6\text{degrees} below zero

−6 degrees

a temperature of 14degrees14\text{degrees} below zero

an elevation of 40feet40\text{feet} below sea level

−40 feet

an elevation of 65feet65\text{feet} below sea level

a football play loss of 12yards12\text{yards}

−12 yards

a football play gain of 4yards4\text{yards}

a stock gain of \text{$3}

$3

a stock loss of \text{$5}

a golf score one above par

+1

a golf score of 33 below par

Everyday Math

Elevation The highest elevation in the United States is Mount McKinley, Alaska, at 20,320feet20,320\text{feet} above sea level. The lowest elevation is Death Valley, California, at 282feet282\text{feet} below sea level. Use integers to write the elevation of:

Mount McKinley
Death Valley
  1. 20,320 feet
  2. −282 feet

Extreme temperatures The highest recorded temperature on Earth is \text{58^\circ Celsius,} recorded in the Sahara Desert in 1922. The lowest recorded temperature is \text{90^\circ } below \text{0^\circ Celsius,} recorded in Antarctica in 1983. Use integers to write the:

highest recorded temperature
lowest recorded temperature

State budgets In June, 2011, the state of Pennsylvania estimated it would have a budget surplus of \text{$540 million.} That same month, Texas estimated it would have a budget deficit of \text{$27 billion.} Use integers to write the budget:

surplus
deficit
  1. $540 million
  2. −$27 billion

College enrollments Across the United States, community college enrollment grew by 1,400,0001,400,000 students from 20072007 to 20102010. In California, community college enrollment declined by 110,171110,171 students from 20092009 to 20102010. Use integers to write the change in enrollment:

growth
decline

Writing Exercises

Give an example of a negative number from your life experience.

Sample answer: I have experienced negative temperatures.

What are the three uses of the "−" sign in algebra? Explain how they differ.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

Practice Makes Perfect

Model Addition of Integers

In the following exercises, model the expression to simplify.

7+47+4

This figure shows a row of 11 light pink circles, representing positive counters. They are separated into a group of seven and a group of four.
11

8+58+5

6+(3)-6+\left(-3\right)

This figure shows a row of 9 dark pink circles, representing negative counters. They are separated into a group of six and a group of three.
−9

5+(5)-5+\left(-5\right)

7+5-7+5

This figure shows two rows of circles. The top row shows 7 dark pink circles, representing negative counters. The bottom row shows 5 light pink circles, representing positive counters.
−2

9+6-9+6

8+(7)8+\left(-7\right)

This figure shows two rows of circles. The top row shows 8 light pink circles, representing positive counters. The bottom row shows 7 light pink circles, representing negative counters.
1

9+(4)9+\left(-4\right)

Simplify Expressions with Integers

In the following exercises, simplify each expression.

21+(59)-21+\left(-59\right)

−80

35+(47)-35+\left(-47\right)

48+(16)48+\left(-16\right)

32

34+(19)34+\left(-19\right)

200+65-200+65

−135

150+45-150+45

2+(8)+62+\left(-8\right)+6

0

4+(9)+74+\left(-9\right)+7

14+(12)+4-14+\left(-12\right)+4

−22

17+(18)+6-17+\left(-18\right)+6

135+(110)+83135+\left(-110\right)+83

108

140+(75)+67140+\left(-75\right)+67

32+24+(6)+10-32+24+\left(-6\right)+10

−4

38+27+(8)+12-38+27+\left(-8\right)+12

19+2(3+8)19+2\left(-3+8\right)

29

24+3(5+9)24+3\left(-5+9\right)

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

x+8x+8 when

x=26x=-26
x=95x=-95
ⓐ −18
ⓑ −87

y+9y+9 when

y=29y=-29
y=84y=-84

y+(14)y+\left(-14\right) when

y=33y=-33
y=30y=30
ⓐ −47
ⓑ 16

x+(21)x+\left(-21\right) when

x=27x=-27
x=44x=44

When a=7a=-7, evaluate:

a+3a+3
a+3-a+3
ⓐ −4
ⓑ 10

When b=11b=-11, evaluate:

b+6b+6
b+6-b+6

When c=9c=-9, evaluate:

c+(4)c+\left(-4\right)
c+(4)-c+\left(-4\right)
ⓐ −13
ⓑ 5

When d=8d=-8, evaluate:

d+(9)d+\left(-9\right)
d+(9)-d+\left(-9\right)

m+nm+n when, m=15m=-15 , n=7n=7

−8

p+qp+q when, p=9p=-9 , q=17q=17

r3sr - 3s when, r=16r=16 , s=2s=2

10

2t+u2t+u when, t=6t=-6 , u=5u=-5

(a+b)2{\left(a+b\right)}^{2} when, a=7a=-7 , b=15b=15

64

(c+d)2{\left(c+d\right)}^{2} when, c=5c=-5 , d=14d=14

(x+y)2{\left(x+y\right)}^{2} when, x=3x=-3 , y=14y=14

121

(y+z)2{\left(y+z\right)}^{2} when, y=3y=-3 , z=15z=15

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

The sum of 14-14 and 55

−14 + 5 = −9

The sum of 22-22 and 99

88 more than 2-2

−2 + 8 = 6

55 more than 1-1

10-10 added to 15-15

−15 + (−10) = −25

6-6 added to 20-20

66 more than the sum of 1-1 and 12-12

[−1 + (−12)] + 6 = −7

33 more than the sum of 2-2 and 8-8

the sum of 1010 and 19-19, increased by 44

[10 + (−19)] + 4 = −5

the sum of 1212 and 15-15, increased by 11

Add Integers in Applications

In the following exercises, solve.

Temperature The temperature in St. Paul, Minnesota was -19\text{^\circ F} at sunrise. By noon the temperature had risen \text{26^\circ F.} What was the temperature at noon?

7°F

Temperature The temperature in Chicago was -15\text{^\circ F} at 6 am. By afternoon the temperature had risen \text{28^\circ F.} What was the afternoon temperature?

Credit Cards Lupe owes \text{$73} on her credit card. Then she charges \text{$45} more. What is the new balance?

−$118

Credit Cards Frank owes \text{$212} on his credit card. Then he charges \text{$105} more. What is the new balance?

Weight Loss Angie lost 3 pounds\text{3 pounds} the first week of her diet. Over the next three weeks, she lost 2 pounds,\text{2 pounds,} gained 1 pound,\text{1 pound,} and then lost 4 pounds.\text{4 pounds.} What was the change in her weight over the four weeks?

−8 pounds

Weight Loss April lost 5 pounds\text{5 pounds} the first week of her diet. Over the next three weeks, she lost 3 pounds,\text{3 pounds,} gained 2 pounds,\text{2 pounds,} and then lost 1 pound.\text{1 pound.} What was the change in her weight over the four weeks?

Football The Rams took possession of the football on their own 35-yard line.\text{35-yard line.} In the next three plays, they lost 12 yards,\text{12 yards,} gained 8 yards,\text{8 yards,} then lost 6 yards.\text{6 yards.} On what yard line was the ball at the end of those three plays?

25-yard line

Football The Cowboys began with the ball on their own 20-yard line.\text{20-yard line.} They gained 15 yards,\text{15 yards,} lost 3 yards\text{3 yards} and then gained 6 yards\text{6 yards} on the next three plays. Where was the ball at the end of these plays?

Calories Lisbeth walked from her house to get a frozen yogurt, and then she walked home. By walking for a total of 20 minutes,\text{20 minutes,} she burned 90 calories.\text{90 calories.} The frozen yogurt she ate was 110 calories.\text{110 calories.} What was her total calorie gain or loss?

20 calories

Calories Ozzie rode his bike for 30 minutes,\text{30 minutes,} burning 168 calories.\text{168 calories.} Then he had a 140-calorie\text{140-calorie} iced blended mocha. Represent the change in calories as an integer.

Everyday Math

Stock Market The week of September 15, 2008, was one of the most volatile weeks ever for the U.S. stock market. The change in the Dow Jones Industrial Average each day was:

Monday504Tuesday+142Wednesday449Thursday+410Friday+369\begin{array}{cccccc}\text{Monday}\hfill & -504\hfill & \text{Tuesday}\hfill & +142\hfill & \text{Wednesday}\hfill & -449\hfill \\ \text{Thursday}\hfill & +410\hfill & \text{Friday}\hfill & +369\hfill & \end{array}

What was the overall change for the week?

−32

Stock Market During the week of June 22, 2009, the change in the Dow Jones Industrial Average each day was:

Monday201Tuesday16Wednesday23Thursday+172Friday34\begin{array}{cccccc}\text{Monday}\hfill & -201\hfill & \text{Tuesday}\hfill & -16\hfill & \text{Wednesday}\hfill & -23\hfill \\ \text{Thursday}\hfill & +172\hfill & \text{Friday}\hfill & -34\hfill & \end{array}

What was the overall change for the week?

Writing Exercises

Explain why the sum of 8-8 and 2\text{2} is negative, but the sum of 8\text{8} and 2-2 and is positive.

Sample answer: In the first case, there are more negatives so the sum is negative. In the second case, there are more positives so the sum is positive.

Give an example from your life experience of adding two negative numbers.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

After reviewing this checklist, what will you do to become confident for all objectives?

Practice Makes Perfect

Model Subtraction of Integers

In the following exercises, model each expression and simplify.

828 - 2

This figure shows a row of 8 light pink circles, representing positive counters. The first 2 are circles and are separated from the last 6.
6

939 - 3

5(1)-5-\left(-1\right)

This figure ishows a row of 5 dark pink circles. The first one is circled.
−4

6(4)-6-\left(-4\right)

54-5 - 4

This figure has a row of 9 dark pink circles representing negative counters. The first 5 are separated from the last 4. Below the last 4 is a row of 4 light pink circles, representing positive counters. These four positive counters are circled.
−9

72-7 - 2

8(4)8-\left(-4\right)

This figure has a row of 12 light pink circles, representing positive counters. The first 8 are separated from the last 4. Below the last 4 is a row of 4 dark pink circles, representing negative counters. These four negative counters are circled.
12

7(3)7-\left(-3\right)

Simplify Expressions with Integers

In the following exercises, simplify each expression.

15615 - 6
15+(6)15+\left(-6\right)
9
9
12912 - 9
12+(9)12+\left(-9\right)
442844 - 28
44+(28)44+\left(-28\right)
16
16
351635 - 16
35+(16)35+\left(-16\right)
8(9)8-\left(-9\right)
8+98+9
17
17
  1. 4(4)4-\left(-4\right)
  2. 4+44+4
  1. 27(18)27-\left(-18\right)
  2. 27+1827+18
  1. 45
  2. 45
  1. 46(37)46-\left(-37\right)
  2. 46+3746+37

In the following exercises, simplify each expression.

15(12)15-\left(-12\right)

27

14(11)14-\left(-11\right)

10(19)10-\left(-19\right)

29

11(18)11-\left(-18\right)

488748 - 87

−39

456945 - 69

317931 - 79

−48

398139 - 81

3111-31 - 11

−42

3218-32 - 18

1742-17 - 42

−59

1946-19 - 46

103(52)-103-\left(-52\right)

−51

105(68)-105-\left(-68\right)

45(54)-45-\left(-54\right)

9

58(67)-58-\left(-67\right)

8378 - 3 - 7

−2

9659 - 6 - 5

54+7-5 - 4+7

−2

38+4-3 - 8+4

14(27)+9-14-\left(-27\right)+9

22

15(28)+5-15-\left(-28\right)+5

71+(10)871+\left(-10\right)-8

53

64+(17)964+\left(-17\right)-9

16(4+1)7-16-\left(-4+1\right)-7

−20

15(6+4)3-15-\left(-6+4\right)-3

(27)(38)\left(2 - 7\right)-\left(3 - 8\right)

0

(18)(29)\left(1 - 8\right)-\left(2 - 9\right)

(68)(24)-\left(6 - 8\right)-\left(2 - 4\right)

4

(45)(78)-\left(4 - 5\right)-\left(7 - 8\right)

25[10(312)]25-\left[10-\left(3 - 12\right)\right]

6

32[5(1520)]32-\left[5-\left(15 - 20\right)\right]

6343726\cdot 3 - 4\cdot 3 - 7\cdot 2

−8

5782495\cdot 7 - 8\cdot 2 - 4\cdot 9

5262{5}^{2}-{6}^{2}

−11

6272{6}^{2}-{7}^{2}

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression for the given values.

x6whenx - 6\text{when}

x=3x=3
x=3x=-3
  1. −3
  2. −9

x4whenx - 4\text{when}

x=5x=5
x=5x=-5

5ywhen5-y\text{when}

y=2y=2
y=2y=-2
  1. 3
  2. 7

8ywhen8-y\text{when}

y=3y=3
y=3y=-3

4x215x+1whenx=34{x}^{2}-15x+1\text{when}x=3

−8

5x214x+7whenx=25{x}^{2}-14x+7\text{when}x=2

125x2whenx=6-12 - 5{x}^{2}\text{when}x=6

−192

194x2whenx=5-19 - 4{x}^{2}\text{when}x=5

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

  1. The difference of 33 and 10-10
  2. Subtract 20-20 from 4545
  1. −3 − (−10) = 13
  2. 45 − (−20) = 65
  1. The difference of 88 and 12-12
  2. Subtract 13-13 from 5050
  1. The difference of 6-6 and 99
  2. Subtract 12-12 from 16-16
  1. −6 − 9 = −15
  2. −16 − (−12) = −4
  1. The difference of 8-8 and 99
  2. Subtract 15-15 from 19-19
  1. 88 less than 17-17
  2. 24-24 minus 3737
  1. −17 − 8 = −25
  2. −24 − 37 = −61
  1. 55 less than 14-14
  2. 13-13 minus 4242
  1. 2121 less than 66
  2. 3131 subtracted from 19-19
  1. 6 − 21 = −15
  2. −19 − 31 = −50
  1. 3434 less than 77
  2. 2929 subtracted from 50-50

Subtract Integers in Applications

In the following exercises, solve the following applications.

Temperature One morning, the temperature in Urbana, Illinois, was \text{28^\circ Fahrenheit.} By evening, the temperature had dropped \text{38^\circ Fahrenheit.} What was the temperature that evening?

−10°

Temperature On Thursday, the temperature in Spincich Lake, Michigan, was \text{22^\circ Fahrenheit.} By Friday, the temperature had dropped \text{35^\circ Fahrenheit.} What was the temperature on Friday?

Temperature On January 15, the high temperature in Anaheim, California, was \text{84^\circ Fahrenheit.} That same day, the high temperature in Embarrass, Minnesota was \text{-12^\circ Fahrenheit.} What was the difference between the temperature in Anaheim and the temperature in Embarrass?

96°

Temperature On January 21, the high temperature in Palm Springs, California, was \text{89^\circ ,} and the high temperature in Whitefield, New Hampshire was \text{-31^\circ }. What was the difference between the temperature in Palm Springs and the temperature in Whitefield?

Football At the first down, the Warriors football team had the ball on their 30-yard line.\text{30-yard line.} On the next three downs, they gained 2 yards,\text{2 yards,} lost 7 yards,\text{7 yards,} and lost 4 yards.\text{4 yards.} What was the yard line at the end of the third down?

21-yard line

Football At the first down, the Barons football team had the ball on their 20-yard line.\text{20-yard line.} On the next three downs, they lost 8 yards,\text{8 yards,} gained 5 yards,\text{5 yards,} and lost 6 yards.\text{6 yards.} What was the yard line at the end of the third down?

Checking Account John has \text{$148} in his checking account. He writes a check for \text{$83.} What is the new balance in his checking account?

$65

Checking Account Ellie has \text{$426} in her checking account. She writes a check for \text{$152.} What is the new balance in her checking account?

Checking Account Gina has \text{$210} in her checking account. She writes a check for \text{$250.} What is the new balance in her checking account?

−$40

Checking Account Frank has \text{$94} in his checking account. He writes a check for \text{$110.} What is the new balance in his checking account?

Checking Account Bill has a balance of \text{-$14} in his checking account. He deposits \text{$40} to the account. What is the new balance?

$26

Checking Account Patty has a balance of \text{-$23} in her checking account. She deposits \text{$80} to the account. What is the new balance?

Everyday Math

Camping Rene is on an Alpine hike. The temperature is -\mathbf{\text{7}}\mathbf{\text{^\circ }}. Rene’s sleeping bag is rated "comfortable to -\mathbf{\text{20}}\text{^\circ ".} How much can the temperature change before it is too cold for Rene’s sleeping bag?

13°

Scuba Diving Shelly’s scuba watch is guaranteed to be watertight to 100feet-100\text{feet}. She is diving at 45feet-45\text{feet} on the face of an underwater canyon. By how many feet can she change her depth before her watch is no longer guaranteed?

Writing Exercises

Explain why the difference of 99 and 6-6 is 1515.

Sample answer: On a number line, 9 is 15 units away from −6.

Why is the result of subtracting 3(4)3-\left(-4\right) the same as the result of adding 3+4?3+4?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Practice Makes Perfect

Multiply Integers

In the following exercises, multiply each pair of integers.

48-4\cdot 8

−32

39-3\cdot 9

5(7)-5\left(7\right)

−35

8(6)-8\left(6\right)

18(2)-18\left(-2\right)

36

10(6)-10\left(-6\right)

9(7)9\left(-7\right)

−63

13(5)13\left(-5\right)

16-1\cdot 6

−6

13-1\cdot 3

1(14)-1\left(-14\right)

14

1(19)-1\left(-19\right)

Divide Integers

In the following exercises, divide.

24÷6-24\div 6

−4

28÷7-28\div 7

56÷(7)56\div \left(-7\right)

−8

35÷(7)35\div \left(-7\right)

52÷(4)-52\div \left(-4\right)

13

84÷(6)-84\div \left(-6\right)

180÷15-180\div 15

−12

192÷12-192\div 12

49÷(1)49\div \left(-1\right)

−49

62÷(1)62\div \left(-1\right)

Simplify Expressions with Integers

In the following exercises, simplify each expression.

5(6)+7(2)35\left(-6\right)+7\left(-2\right)-3

−47

8(4)+5(4)68\left(-4\right)+5\left(-4\right)-6

8(2)3(9)-8\left(-2\right)-3\left(-9\right)

43

7(4)5(3)-7\left(-4\right)-5\left(-3\right)

(5)3{\left(-5\right)}^{3}

−125

(4)3{\left(-4\right)}^{3}

(2)6{\left(-2\right)}^{6}

64

(3)5{\left(-3\right)}^{5}

42-{4}^{2}

−16

62-{6}^{2}

3(5)(6)-3\left(-5\right)\left(6\right)

90

4(6)(3)-4\left(-6\right)\left(3\right)

4211-4\cdot 2\cdot 11

−88

5310-5\cdot 3\cdot 10

(811)(912)\left(8 - 11\right)\left(9 - 12\right)

9

(611)(813)\left(6 - 11\right)\left(8 - 13\right)

263(27)26 - 3\left(2 - 7\right)

41

232(46)23 - 2\left(4 - 6\right)

10(4)÷(8)-10\left(-4\right)\div \left(-8\right)

−5

8(6)÷(4)-8\left(-6\right)\div \left(-4\right)

65÷(5)+(28)÷(7)65\div \left(-5\right)+\left(-28\right)\div \left(-7\right)

−9

52÷(4)+(32)÷(8)52\div \left(-4\right)+\left(-32\right)\div \left(-8\right)

92[38(2)]9 - 2\left[3 - 8\left(-2\right)\right]

−29

113[74(2)]11 - 3\left[7 - 4\left(-2\right)\right]

(3)224÷(82){\left(-3\right)}^{2}-24\div \left(8 - 2\right)

5

(4)232÷(124){\left(-4\right)}^{2}-32\div \left(12 - 4\right)

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

2x+17when-2x+17\text{when}

x=8x=8
x=8x=-8
1
33

5y+14when-5y+14\text{when}

y=9y=9
y=9y=-9

103mwhen10 - 3m\text{when}

m=5m=5
m=5m=-5
−5
25

184nwhen18 - 4n\text{when}

n=3n=3
n=3n=-3

p25p+5whenp=1{p}^{2}-5p+5\text{when}p=-1

8

q22q+9whenq=2{q}^{2}-2q+9\text{when}q=-2

2w23w+7whenw=22{w}^{2}-3w+7\text{when}w=-2

21

3u24u+5whenu=33{u}^{2}-4u+5\text{when}u=-3

6x5y+15whenx=3andy=16x - 5y+15\text{when}x=3\text{and}y=-1

38

3p2q+9whenp=8andq=23p - 2q+9\text{when}p=8\text{and}q=-2

9a2b8whena=6andb=39a - 2b - 8\text{when}a=-6\text{and}b=-3

−56

7m4n2whenm=4andn=97m - 4n - 2\text{when}m=-4\text{and}n=-9

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

The product of 3-3 and 15

−3·15 = −45

The product of 4-4 and 1616

The quotient of 60-60 and 20-20

−60 ÷ (−20) = 3

The quotient of 40-40 and 20-20

The quotient of 6-6 and the sum of aa and bb

6a+b\frac{-6}{a+b}

The quotient of 7-7 and the sum of mm and nn

The product of 10-10 and the difference of pandqp\text{and}q

−10 (pq)

The product of 13-13 and the difference of canddc\text{and}d

Everyday Math

Stock market Javier owns 300300 shares of stock in one company. On Tuesday, the stock price dropped \text{$12} per share. What was the total effect on Javier’s portfolio?

−$3,600

Weight loss In the first week of a diet program, eight women lost an average of 3 pounds\text{3 pounds} each. What was the total weight change for the eight women?

Writing Exercises

In your own words, state the rules for multiplying two integers.

Sample answer: Multiplying two integers with the same sign results in a positive product. Multiplying two integers with different signs results in a negative product.

In your own words, state the rules for dividing two integers.

Why is 24(2)4?{-2}^{4}\ne {\left(-2\right)}^{4}?

Sample answer: In one expression the base is positive and then we take the opposite, but in the other the base is negative.

Why is 42(4)2?{-4}^{2}\ne {\left(-4\right)}^{2}?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Chapter Review Exercises

Use the Language of Algebra

Use Variables and Algebraic Symbols

In the following exercises, translate from algebra to English.

383\cdot 8

the product of 3 and 8

12x12-x

24÷624\div 6

the quotient of 24 and 6

9+2a9+2a

504750\ge 47

50 is greater than or equal to 47

3y<153y<15

n+4=13n+4=13

The sum of n and 4 is equal to 13

32k=732-k=7

Identify Expressions and Equations

In the following exercises, determine if each is an expression or equation.

5+u=845+u=84

equation

366s36 - 6s

4y114y - 11

expression

10x=12010x=120

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

2222\cdot 2\cdot 2

23

aaaaaa\cdot a\cdot a\cdot a\cdot a

xxxxxxx\cdot x\cdot x\cdot x\cdot x\cdot x

x6

10101010\cdot 10\cdot 10

In the following exercises, write in expanded form.

84{8}^{4}

8 ⋅ 8 ⋅ 8 ⋅ 8

36{3}^{6}

y5{y}^{5}

yyyyy

n4{n}^{4}

In the following exercises, simplify each expression.

34{3}^{4}

81

106{10}^{6}

27{2}^{7}

128

43{4}^{3}

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

10+2510+2\cdot 5

20

(10+2)5\left(10+2\right)\cdot 5

(30+6)÷2\left(30+6\right)\div 2

18

30+6÷230+6\div 2

72+52{7}^{2}+{5}^{2}

74

(7+5)2{\left(7+5\right)}^{2}

4+3(101)4+3\left(10 - 1\right)

31

(4+3)(101)\left(4+3\right)\left(10 - 1\right)

Evaluate, Simplify, and Translate Expressions

Evaluate an Expression

In the following exercises, evaluate the following expressions.

9x5whenx=79x - 5\text{when}x=7

58

y3wheny=5{y}^{3}\text{when}y=5

3a4b3a - 4b when a=10,b=1a=10,b=1

26

bhwhenb=7,h=8bh\text{when}b=7,h=8

Identify Terms, Coefficients and Like Terms

In the following exercises, identify the terms in each expression.

12n2+3n+112{n}^{2}+3n+1

12n2,3n, 1

4x3+11x+34{x}^{3}+11x+3

In the following exercises, identify the coefficient of each term.

6y6y

6

13x213{x}^{2}

In the following exercises, identify the like terms.

5x2,3,5y2,3x,x,45{x}^{2},3,5{y}^{2},3x,x,4

3, 4, and 3x, x

8,8r2,8r,3r,r2,3s8,8{r}^{2},\text{8}r,3r,{r}^{2},3s

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the following expressions by combining like terms.

15a+9a15a+9a

24a

12y+3y+y12y+3y+y

4x+7x+3x4x+7x+3x

14x

6+5c+36+5c+3

8n+2+4n+98n+2+4n+9

12n + 11

19p+5+4p1+3p19p+5+4p - 1+3p

7y2+2y+11+3y287{y}^{2}+2y+11+3{y}^{2}-8

10y2 + 2y + 3

13x2x+6+5x2+9x13{x}^{2}-x+6+5{x}^{2}+9x

Translate English Phrases to Algebraic Expressions

In the following exercises, translate the following phrases into algebraic expressions.

the difference of xx and 66

x − 6

the sum of 1010 and twice aa

the product of 3n3n and 99

3n ⋅ 9

the quotient of ss and 44

55 times the sum of yy and 11

5(y + 1)

1010 less than the product of 55 and zz

Jack bought a sandwich and a coffee. The cost of the sandwich was \text{$3} more than the cost of the coffee. Call the cost of the coffee cc. Write an expression for the cost of the sandwich.

c + 3

The number of poetry books on Brianna’s bookshelf is 55 less than twice the number of novels. Call the number of novels nn. Write an expression for the number of poetry books.

Solve Equations Using the Subtraction and Addition Properties of Equality

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each number is a solution to the equation.

y+16=40y+16=40

2424
5656
yes
no

d6=21d - 6=21

1515
2727

4n+12=364n+12=36

66
1212
yes
no

20q10=7020q - 10=70

33
44

15x5=10x+4515x - 5=10x+45

22
1010
no
yes

22p6=18p+8622p - 6=18p+86

44
2323

Model the Subtraction Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters and then solve the equation using the subtraction property of equality.

This image is divided into two parts: the first part shows an envelope and 3 blue counters and the next to it, the second part shows five counters.

x + 3 = 5; x = 2

This image is divided into two parts: the first part shows an envelope and 4 blue counters and next to it, the second part shows 9 counters.

Solve Equations using the Subtraction Property of Equality

In the following exercises, solve each equation using the subtraction property of equality.

c+8=14c+8=14

6

v+8=150v+8=150

23=x+1223=x+12

11

376=n+265376=n+265

Solve Equations using the Addition Property of Equality

In the following exercises, solve each equation using the addition property of equality.

y7=16y - 7=16

23

k42=113k - 42=113

19=p1519=p - 15

34

501=u399501=u - 399

Translate English Sentences to Algebraic Equations

In the following exercises, translate each English sentence into an algebraic equation.

The sum of 77 and 3333 is equal to 4040.

7 + 33 = 44

The difference of 1515 and 33 is equal to 1212.

The product of 44 and 88 is equal to 3232.

4 ⋅ 8 = 32

The quotient of 6363 and 99 is equal to 77.

Twice the difference of nn and 33 gives 7676.

2(n − 3) = 76

The sum of five times yy and 44 is 8989.

Translate to an Equation and Solve

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

Eight more than xx is equal to 3535.

x + 8 = 35; x = 27

2121 less than aa is 1111.

The difference of qq and 1818 is 5757.

q − 18 = 57; q = 75

The sum of mm and 125125 is 240240.

Mixed Practice

In the following exercises, solve each equation.

h15=27h - 15=27

h = 42

k11=34k - 11=34

z+52=85z+52=85

z = 33

x+93=114x+93=114

27=q+1927=q+19

q = 8

38=p+1938=p+19

31=v2531=v - 25

v = 56

38=u1638=u - 16

Find Multiples and Factors

Identify Multiples of Numbers

In the following exercises, list all the multiples less than 5050 for each of the following.

33

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48

22

88

8, 16, 24, 32, 40, 48

1010

Use Common Divisibility Tests

In the following exercises, using the divisibility tests, determine whether each number is divisible by 2,by3,by5,by6,and by102,\text{by}3,\text{by}5,\text{by}6,\text{and by}10.

9696

2, 3, 6

250250

420420

2, 3, 5, 6, 10

625625

Find All the Factors of a Number

In the following exercises, find all the factors of each number.

3030

1, 2, 3, 5, 6, 10, 15, 30

7070

180180

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180

378378

Identify Prime and Composite Numbers

In the following exercises, identify each number as prime or composite.

1919

prime

5151

121121

composite

219219

Prime Factorization and the Least Common Multiple

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number.

8484

2 ⋅ 2 ⋅ 3 ⋅ 7

165165

350350

2 ⋅ 5 ⋅ 5 ⋅ 7

572572

Find the Least Common Multiple of Two Numbers

In the following exercises, find the least common multiple of each pair of numbers.

9,159,15

45

12,2012,20

25,3525,35

350

18,4018,40

Everyday Math

Describe how you have used two topics from The Language of Algebra chapter in your life outside of your math class during the past month.

Answers will vary

Chapter Practice Test

In the following exercises, translate from an algebraic equation to English phrases.

646\cdot 4

15x15-x

fifteen minus x

In the following exercises, identify each as an expression or equation.

58+105\cdot 8+10

x+6=9x+6=9

equation

311=333\cdot 11=33

Write nnnnnnn\cdot n\cdot n\cdot n\cdot n\cdot n in exponential form.
Write 35{3}^{5} in expanded form and then simplify.
n6
3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 243

In the following exercises, simplify, using the order of operations.

4+354+3\cdot 5

(8+1)4\left(8+1\right)\cdot 4

36

1+6(31)1+6\left(3 - 1\right)

(8+4)÷3+1\left(8+4\right)\div 3+1

5

(1+4)2{\left(1+4\right)}^{2}

5[2+7(98)]5\left[2+7\left(9 - 8\right)\right]

45

In the following exercises, evaluate each expression.

8x3whenx=48x - 3\text{when}x=4

y3wheny=5{y}^{3}\text{when}y=5

125

6a2bwhena=5,b=76a - 2b\text{when}a=5,b=7

hwwhenh=12,w=3hw\text{when}h=12,w=3

36

Simplify by combining like terms.

6x+8x6x+8x
9m+10+m+39m+10+m+3

In the following exercises, translate each phrase into an algebraic expression.

55 more than xx

x + 5

the quotient of 1212 and yy

three times the difference of aandba\text{and}b

3(ab)

Caroline has 33 fewer earrings on her left ear than on her right ear. Call the number of earrings on her right ear, rr. Write an expression for the number of earrings on her left ear.

In the following exercises, solve each equation.

n6=25n - 6=25

n = 31

x+58=71x+58=71

In the following exercises, translate each English sentence into an algebraic equation and then solve it.

1515 less than yy is 3232.

y − 15 = 32; y = 47

the sum of aa and 129129 is 164164.

List all the multiples of 44, that are less than 5050.

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

Find all the factors of 9090.

Find the prime factorization of 10801080.

23 ⋅ 33 ⋅ 5

Find the LCM (Least Common Multiple) of 2424 and 4040.

Glossary

least common multiple
The smallest number that is a multiple of two numbers is called the least common multiple (LCM).)
prime factorization
The prime factorization of a number is the product of prime numbers that equals the number.
 

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