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Prealgebra

Chapter 3

PrealgebraChapter 3
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Table of contents
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Chapter Review
      1. Key Terms
      2. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Chapter Review
      1. Key Terms
      2. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

Try It

3.3
  1. ⓐ >
  2. ⓑ <
  3. ⓒ >
  4. ⓓ >
3.4
  1. ⓐ <
  2. ⓑ >
  3. ⓒ <
  4. ⓓ >
3.5
  1. ⓐ −4
  2. ⓑ 3
3.6
  1. ⓐ −8
  2. ⓑ 5
3.7

1

3.8

5

3.9
  1. ⓐ −4
  2. ⓑ 4
3.10
  1. ⓐ −11
  2. ⓑ 11
3.11
  1. ⓐ 12
  2. ⓑ −28
3.12
  1. ⓐ 9
  2. ⓑ −37
3.13
  1. ⓐ 17
  2. ⓑ 39
  3. ⓒ −22
  4. ⓓ −11
3.14
  1. ⓐ 23
  2. ⓑ 21
  3. ⓒ −37
  4. ⓓ −49
3.15
  1. ⓐ >
  2. ⓑ >
  3. ⓒ <
  4. ⓓ =
3.16
  1. ⓐ>
  2. ⓑ =
  3. ⓒ >
  4. ⓓ <
3.17
  1. ⓐ 3
  2. ⓑ 18
3.18
  1. ⓐ 11
  2. ⓑ 63
3.19

2

3.20

3

3.21

16

3.22

9

3.23
  1. ⓐ −9
  2. ⓑ 15
  3. ⓒ −20
  4. ⓓ 11−(−4)
3.24
  1. ⓐ 19
  2. ⓑ −22
  3. ⓒ −9
  4. ⓓ −8−(−5)
3.25

5 yards

3.26

−30 feet

3.27


This figure has six pink circles in a row, representing positive counters. The first two circles are separated from the following four circles.


6

3.28


This figure has seven pink circles in a row, representing positive counters. Two circles are separated from the following 5 circles.


7

3.29


This figure shows a row of 6 dark pink circles, representing negative counters. They are grouped by 2 circles followed by 4 circles.


−6

3.30


This figure shows a row of 7 dark pink circles, representing negative counters. They are grouped by 2 circles followed by 5 circles.


−7

3.31


This figure shows two rows of counter circles. The first row has 2 dark pink circles, representing negative counters. The second row has 4 light pink circles, representing positive counters.


−2

3.32


This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 5 dark pink circles, representing negative counters.


−3

3.33


This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 4 dark pink circles, representing negative counters.


2

3.34


This figure shows two rows of counters. The first row shows 2 dark pink circles, representing negative counters. The second row has 5 light pink circles, representing negative counters. The neutral pairs of one positive and one negative counter are circled leaving three positive counters.


3

3.35
  1. ⓐ
    .
  2. ⓑ
    .
  3. ⓒ
    .
  4. ⓓ
    .
3.36
  1. ⓐ
    .
  2. ⓑ
    .
  3. ⓒ
    .
  4. ⓓ
    .
3.37
  1. ⓐ −17
  2. ⓑ 57
3.38
  1. ⓐ −46
  2. ⓑ 26
3.39

−50

3.40

−70

3.41

13

3.42

0

3.43
  1. ⓐ 2
  2. ⓑ −12
3.44
  1. ⓐ 2
  2. ⓑ −1
3.45
  1. ⓐ −6
  2. ⓑ 10
3.46
  1. ⓐ −1
  2. ⓑ 17
3.47

9

3.48

13

3.49

196

3.50

8

3.51

−7 + 4 = −3

3.52

−8 + (−6) = −14

3.53

[9 + (−16)] + 4 = −3

3.54

[−8 + (−12)] + 7 = −13

3.55

4 degrees Celsius

3.56

−33 feet

3.57

14-yard line

3.58

37-yard line

3.59


This figure shows a row of 6 light pink circles, representing positive counters. The first four are circled.


2

3.60


This figure shows a row of 7 light pink circles representing positive counters. The first four counters are circled.


3

3.61


This figure shows a row of 6 dark pink circles, representing negative counters. The last four counters are circled.


−2

3.62


This figure is a row of 7 dark pink circles representing negative counters. The first four counters are circled.


−3

3.63


This figure shows a row of 10 dark pink circles, representing negative counters. The first six counters are separated from the last four. Below the dark pink circles are four light pink circles, representing positive counters. These four positive counters are circled.


−10

3.64


This figure shows a row of 11 dark pink circles, representing negative counters. The first seven counters are separated from the last four. Below the dark pink circles are four light pink circles, representing positive counters. These four positive counters are circled.


−11

3.65


This figure shows a row of 10 light pink circles, representing positive counters. The first six counters are separated from the last four. Below the light pink circles are four dark pink circles, representing negative counters. These four negative counters are circled.


10

3.66


This figure shows a row of 11 light pink circles, representing positive counters. The first seven counters are separated from the last four. Below the light pink circles are four dark pink circles, representing negative counters. These four negative counters are circled.


11

3.67

ⓐ

.


 ⓑ

.


 ⓒ

.


 ⓓ

.
3.68

ⓐ

.


 ⓑ

.


 ⓒ

.


 ⓓ

.
3.69

ⓐ

This figure shows a row of 9 light pink circles, representing positive counters. The first seven are separated from the last two. The entire row is circled. Below the last two light pink circles is a row of two dark pink circles, representing negative counters.


−2
ⓑ

This figure shows a row of 9 dark pink circles, representing negative counters. The first five are separated from the last four. The entire row is circled. Below the last four dark pink circles is a row of four light pink circles, representing positive counters.


4

3.70

ⓐ

This figure shows a row of seven light pink circles, representing positive counters. The first four are separated from the last three. The entire row is circled. Below the last three light pink circles is a row of three dark pink circles, representing negative counters.


−3
 ⓑ

This figure shows a row of 10 dark pink circles, representing negative counters. The first seven are separated from the last three. The entire row is circled. Below the last three dark pink circles is a row of three light pink circles, representing positive counters.


3

3.71
  1. ⓐ 8, 8
  2. ⓑ −18, −18
3.72
  1. ⓐ 8, 8
  2. ⓑ −22, −22
3.73
  1. ⓐ 19, 19
  2. ⓑ −4, −4
3.74
  1. ⓐ 23, 23
  2. ⓑ 3, 3
3.75

−29

3.76

−26

3.77

3

3.78

13

3.79

−69

3.80

−47

3.81
  1. ⓐ −2
  2. ⓑ −15
3.82
  1. ⓐ −2
  2. ⓑ −7
3.83
  1. ⓐ −2
  2. ⓑ 36
3.84
  1. ⓐ −19
  2. ⓑ 9
3.85
  1. ⓐ −14 − (−23) = 37
  2. ⓑ −17 − 21 = −38
3.86
  1. ⓐ 11 − (−19) = 30
  2. ⓑ −11 − 18 = −29
3.87

45 degrees Fahrenheit

3.88

9 degrees Fahrenheit

3.89

10,103 feet

3.90

233 feet

3.91
  1. ⓐ $48
  2. ⓑ −$2
  3. ⓒ $18
3.92
  1. ⓐ −$54
  2. ⓑ No, −$5
3.93
  1. ⓐ −48
  2. ⓑ 28
  3. ⓒ −63
  4. ⓓ 60
3.94
  1. ⓐ −56
  2. ⓑ 54
  3. ⓒ −28
  4. ⓓ 39
3.95
  1. ⓐ −9
  2. ⓑ 17
3.96
  1. ⓐ −8
  2. ⓑ 16
3.97
  1. ⓐ −7
  2. ⓑ 39
3.98
  1. ⓐ −9
  2. ⓑ 23
3.99
  1. ⓐ −6
  2. ⓑ 36
3.100
  1. ⓐ −28
  2. ⓑ 52
3.101

−63

3.102

−84

3.103
  1. ⓐ 81
  2. ⓑ −81
3.104
  1. ⓐ 49
  2. ⓑ −49
3.105

29

3.106

52

3.109

21

3.111

39

3.112

13

3.113

−8

3.114

19

3.115

−5 (12) = −60

3.116

8 (−13) = −104

3.117

−63 ÷ −9 = 7

3.118

−72 ÷ −9 = 8

3.119
  1. ⓐ no
  2. ⓑ no
  3. ⓒ yes
3.120
  1. ⓐ no
  2. ⓑ no
  3. ⓒ yes
3.121

−4

3.122

−19

3.123

−6

3.124

−4

3.125

4x = 12; x = 3

3.126

3x = 6; x = 2

3.128

11

3.129

−12

3.130

−9

3.131

x + 7 = −2; x = −9

3.132

y + 11 = 2; y = −9

3.133

p − 2 = −4; p = −2

3.134

q − 7 = −3; q = 4

3.135

132 = −12y; y = −11

3.136

117 = −13z; z = −9

Section 3.1 Exercises

1.


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.
3.


This figure is a number line. Negative 8 is labeled a, negative 6 is labeled c, and 5 is labeled b.
5.
  1. ⓐ >
  2. ⓑ <
  3. ⓒ <
  4. ⓓ >
7.
  1. ⓐ <
  2. ⓑ >
  3. ⓒ <
  4. ⓓ >
9.
  1. ⓐ −2
  2. ⓑ 6
11.
  1. ⓐ 8
  2. ⓑ −1
13.

4

15.

15

17.
  1. ⓐ −3
  2. ⓑ 3
19.
  1. ⓐ −12;
  2. ⓑ 12
21.
  1. ⓐ 7
  2. ⓑ 25
  3. ⓒ 0
23.
  1. ⓐ 32
  2. ⓑ 18
  3. ⓒ 16
25.
  1. ⓐ 28
  2. ⓑ 15
27.
  1. ⓐ −19
  2. ⓑ −33
29.
  1. ⓐ <
  2. ⓑ =
31.
  1. ⓐ >
  2. ⓑ >
33.

4

35.

56

37.

0

39.

8

41.

80

43.
  1. ⓐ −8
  2. ⓑ −(−6), or 6
  3. ⓒ −3
  4. ⓓ 4−(−3)
45.
  1. ⓐ −20
  2. ⓑ −(−5), or 5
  3. ⓒ −12
  4. ⓓ 18−(−7)
47.

−6 degrees

49.

−40 feet

51.

−12 yards

53.

$3

55.

+1

57.
  1. ⓐ 20,320 feet
  2. ⓑ −282 feet
59.
  1. ⓐ $540 million
  2. ⓑ −$27 billion
61.

Sample answer: I have experienced negative temperatures.

Section 3.2 Exercises

63.


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

65.


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

67.


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

69.


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

71.

−80

73.

32

75.

−135

77.

0

79.

−22

81.

108

83.

−4

85.

29

87.
  1. ⓐ −18
  2. ⓑ −87
89.
  1. ⓐ −47
  2. ⓑ 16
91.
  1. ⓐ −4
  2. ⓑ 10
93.
  1. ⓐ −13
  2. ⓑ 5
95.

−8

97.

10

99.

64

101.

121

103.

−14 + 5 = −9

105.

−2 + 8 = 6

107.

−15 + (−10) = −25

109.

[−1 + (−12)] + 6 = −7

111.

[10 + (−19)] + 4 = −5

113.

7°F

115.

−$118

117.

−8 pounds

119.

25-yard line

121.

20 calories

123.

−32

125.

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.

Section 3.3 Exercises

127.


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

129.


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


−4

131.


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

133.


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

135.
  1. ⓐ 9
  2. ⓑ 9
137.
  1. ⓐ 16
  2. ⓑ 16
139.
  1. ⓐ 17
  2. ⓑ 17
141.
  1. ⓐ 45
  2. ⓑ 45
143.

27

145.

29

147.

−39

149.

−48

151.

−42

153.

−59

155.

−51

157.

9

159.

−2

161.

−2

163.

22

165.

53

167.

−20

169.

0

171.

4

173.

6

175.

5.2

177.

−11

179.
  1. ⓐ −3
  2. ⓑ −9
181.
  1. ⓐ 3
  2. ⓑ 7
183.

−8

185.

−192

187.
  1. ⓐ −3 − (−10) = 13
  2. ⓑ 45 − (−20) = 65
189.
  1. ⓐ −6 − 9 = −15
  2. ⓑ −16 − (−12) = −4
191.
  1. ⓐ −17 − 8 = −25
  2. ⓑ −24 − 37 = −61
193.
  1. ⓐ 6 − 21 = −15
  2. ⓑ −19 − 31 = −50
195.

−10°

197.

96°

199.

21-yard line

201.

$65

203.

−$40

205.

$26

207.

13°

209.

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

Section 3.4 Exercises

211.

−32

213.

−35

215.

36

217.

−63

219.

−6

221.

14

223.

−4

225.

−8

227.

13

229.

−12

231.

−49

233.

−47

235.

43

237.

−125

239.

64

241.

−16

243.

90

245.

−88

247.

9

249.

41

251.

−5

253.

−9

255.

−29

257.

5

259.
  1. ⓐ 1
  2. ⓑ 33
261.
  1. ⓐ −5
  2. ⓑ 25
263.

8

265.

21

267.

38

269.

−56

271.

−3·15 = −45

273.

−60 ÷ (−20) = 3

275.

−6 a + b −6 a + b

277.

−10 (p − q)

279.

−$3,600

281.

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.

283.

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

Section 3.5 Exercises

285.
  1. ⓐ no
  2. ⓑ no
  3. ⓒ yes
287.
  1. ⓐ no
  2. ⓑ no
  3. ⓒ yes
289.

−7

291.

−17

293.

−4

295.

6

297.

−16

299.

−14

301.

3x = 6; x = 2

303.

2x = 8; x = 4

305.

9

307.

−8

309.

3

311.

−12

313.

20

315.

0

317.

n + 4 = 1; n = −3

319.

8 + p = −3; p = −11

321.

a − 3 = −14; a = −11

323.

−42 = −7x; x = 6

325.

f(−15) = 75; f = 5

327.

−6 + c = 4; c = 10

329.

m − 9 = −4; m = 5

331.
  1. ⓐ 8
  2. ⓑ 5
333.
  1. ⓐ −9
  2. ⓑ 30
335.

20

337.

7

339.

−52

341.

−9

343.

5

345.

−42

347.

17 cookies

349.

Sample answer: It is helpful because it shows how the counters can be divided among the envelopes.

351.

Sample answer: The operation used in the equation is multiplication. The inverse of multiplication is division, not addition.

Review Exercises

353.


This figure is a number line. It is scaled from negative 10 to 10 in increments of 2. There is a point at 5.
355.


This figure is a number line. It is scaled from negative 10 to 10 in increments of 2. There is a point at negative 3.
357.


This figure is a number line. It is scaled from negative 10 to 10 in increments of 2. There is a point at negative 8.
359.

<

361.

>

363.

>

365.

−6

367.

4

369.
  1. ⓐ −8
  2. ⓑ 8
371.
  1. ⓐ −32
  2. ⓑ 32
373.

21

375.

36

377.

0

379.

14

381.

−33

383.

<

385.

=

387.

55; −55

389.

7

391.

54

393.

−1

395.

−16

397.

−3

399.

−10°

401.

10

403.

1

405.

96

407.

−50

409.

−1

411.

21

413.
  1. ⓐ 3
  2. ⓑ −16
415.

−27

417.

−8 + 2 = −6

419.

10 + [−5 + (−6)] = −1

421.

16 degrees

423.


This figure is a row of 6 light pink circles, representing positive counters. The first one is circled.


5

425.


This figure shows 2 rows. The first row shows 7 light pink circles, representing positive counters. The second row shows 5 dark pink circles, representing negative counters. The entire second row is circled.


7

427.

8

429.

−38

431.

−58

433.

−1

435.
  1. ⓐ −2
  2. ⓑ −11
437.

41

439.

−12 − 5 = −17

441.

−2 degrees

443.

−36

445.

121

447.

−7

449.

−8

451.

−45

453.

−9

455.

−81

457.

54

459.

4

461.

−66

463.

−58

465.

−12(6) = −72

467.
  1. ⓐ no
  2. ⓑ yes
  3. ⓒ no
469.

−12

470.

−6

471.

−7

472.

−32

473.

3x = 9; x = 3>

475.

9

477.

4

479.

−6y = −42; y = 7

481.

m + 4 = −48; m = −52

Practice Test

485.
  1. ⓐ <
  2. ⓑ >
487.
  1. ⓐ 7
  2. ⓑ −8
489.

5

491.

−27

493.

11

495.

54

497.

−8

499.

22

501.

39

503.

34

505.

−7 − (−4) = −3

507.

4°F

509.

n = −1

511.

r = 6

513.

y − 8 = −32; y = −24

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