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Intermediate Algebra 2e

Chapter 1

Intermediate Algebra 2eChapter 1
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Table of contents
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Chapter Review
      1. Key Terms
      2. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Chapter Review
      1. Key Terms
      2. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  14. 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
    12. Chapter 12
  15. Index

Try It

1.1

ⓐ yes ⓑ yes ⓒ no ⓓ yes
ⓔ no

1.2

ⓐ no ⓑ yes ⓒ yes ⓓ no
ⓔ no

1.3

2 · 2 · 2 · 2 · 5 2 · 2 · 2 · 2 · 5

1.4

2 · 2 · 3 · 5 2 · 2 · 3 · 5

1.5

36 36

1.6

72 72

1.7

16

1.8

23

1.9

86

1.10

1

1.11

ⓐ 9 ⓑ 64 ⓒ 40

1.12

ⓐ 216 ⓑ 64 ⓒ 185

1.13

10 x 2 + 16 x + 17 10 x 2 + 16 x + 17

1.14

12 y 2 + 9 y + 7 12 y 2 + 9 y + 7

1.15

ⓐ 14x2−1314x2−13 ⓑ 12x÷212x÷2
ⓒ z+13z+13 ⓓ 8x−188x−18

1.16

ⓐ 17y2+1917y2+19 ⓑ 7y7y
ⓒ x+11x+11 ⓓ 11a−1411a−14

1.17

ⓐ 4(p+q)4(p+q) ⓑ 4p+q4p+q

1.18

ⓐ 2x−82x−8 ⓑ 2(x−8)2(x−8)

1.19

w − 7 w − 7

1.20

l − 6 l − 6

1.21

4 q − 8 4 q − 8

1.22

7 n + 3 7 n + 3

1.23

ⓐ == ⓑ >> ⓒ <<
ⓓ ==

1.24

ⓐ >> ⓑ == ⓒ >>
ⓓ <<

1.25

16

1.26

9

1.27

ⓐ −6−6 ⓑ 2 ⓒ −2−2

1.28

ⓐ −7−7 ⓑ 3 ⓒ −3−3

1.29

ⓐ 2 ⓑ −2−2 ⓒ −10−10 ⓓ 10

1.30

ⓐ 3 ⓑ −3−3 ⓒ −11−11 ⓓ 11

1.31

ⓐ 8,88,8 ⓑ −18,−18, −18−18
ⓒ 19,1919,19 ⓓ −4,−4, −4−4

1.32

ⓐ 8,88,8 ⓑ −22,−22, −22−22
ⓒ 23,2323,23 ⓓ 3,33,3

1.33

3

1.34

13

1.35

ⓐ 23 ⓑ 60 ⓒ −63−63 ⓓ −9−9

1.36

ⓐ 39 ⓑ 39 ⓒ −28 ⓓ −7

1.37

ⓐ 81 ⓑ −81−81

1.38

ⓐ 49 ⓑ −49−49

1.39

ⓐ 4 ⓑ 21

1.40

ⓐ 9 ⓑ 6

1.41

31

1.42

67

1.43

( 9 + ( −16 ) ) + 4 ; − 3 ( 9 + ( −16 ) ) + 4 ; − 3

1.44

( −8 + ( −12 ) ) + 7 ; − 13 ( −8 + ( −12 ) ) + 7 ; − 13

1.45

The difference in temperatures was 45 degrees.

1.46

The difference in temperatures was 9 degrees.

1.47

− 23 40 − 23 40

1.48

− 5 8 − 5 8

1.49

−33 a −33 a

1.50

−26 b −26 b

1.51

4 15 4 15

1.52

2 3 2 3

1.53

3 4 b 3 4 b

1.54

4 q 4 q

1.55

79 60 79 60

1.56

103 60 103 60

1.57

ⓐ 27a−323627a−3236 ⓑ 2a32a3

1.58

ⓐ 24k−53024k−530 ⓑ 2k152k15

1.59

4

1.60

2

1.61

1 90 1 90

1.62

272

1.63

2

1.64

2 7 2 7

1.65

− 1 2 − 1 2

1.66

2 3 2 3

1.67

ⓐ 6.586.58 ⓑ 6.66.6 ⓒ 7

1.68

ⓐ 15.21815.218 ⓑ 15.2215.22
ⓒ 15.215.2

1.69

ⓐ −16.49−16.49 ⓑ −0.42−0.42

1.70

ⓐ −23.593−23.593 ⓑ −12.58−12.58

1.71

−27.4815 −27.4815

1.72

−87.6148 −87.6148

1.73

ⓐ 25.8 ⓑ 258 ⓒ 2,580

1.74

ⓐ 142 ⓑ 1,420 ⓒ 14,200

1.75

587.3 587.3

1.76

34.25 34.25

1.77

ⓐ 117500117500 ⓑ −0.875−0.875

1.78

ⓐ 31253125 ⓑ −0.375−0.375

1.79

ⓐ 0.09, 0.87, 0.039 ⓑ 17%, 175%, 8.25%

1.80

ⓐ 0.03, 0.91, 0.083 ⓑ 41%, 225%, 9.25%

1.81

ⓐ 6 ⓑ 13 ⓒ −15−15

1.82

ⓐ 4 ⓑ 14 ⓒ −10−10

1.83

ⓐ 4,494,49 ⓑ −3,4,49−3,4,49
ⓒ −3,0.3–,95,4,49−3,0.3–,95,4,49 ⓓ −2−2
ⓔ −3,−2,0.3–,95,4,49−3,−2,0.3–,95,4,49

1.84

ⓐ 6,1216,121
ⓑ −25,−1,6,121−25,−1,6,121
ⓒ −25,−38,−1,6,121−25,−38,−1,6,121
ⓓ 2.041975...2.041975...
ⓔ −25,−38,−1,6,121,2.041975...−25,−38,−1,6,121,2.041975...

1.87

ⓐ

Figure shows a number line with numbers ranging from 0 to 1. 0.6 is highlighted.

ⓑ

Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted, minus 0.25 is highlighted.
1.88

ⓐ

Figure shows a number line with numbers ranging from 0 to 1. 0.9 is highlighted.

ⓑ

Figure shows a number line with numbers ranging from minus 1.00 to 0.00. Minus 0.74 is highlighted.
1.89

32 r + 29 s 32 r + 29 s

1.90

41 m + 6 n 41 m + 6 n

1.91

1 7 15 1 7 15

1.92

1 2 9 1 2 9

1.93

−48 a −48 a

1.94

−92 x −92 x

1.95

5 49 5 49

1.96

11 25 11 25

1.97

ⓐ 0 ⓑ undefined

1.98

ⓐ 0 ⓑ undefined

1.99

4 x + 8 4 x + 8

1.100

6 x + 42 6 x + 42

1.101

5 y + 3 5 y + 3

1.102

4 n + 9 4 n + 9

1.103

70 + 15 p 70 + 15 p

1.104

4 + 35 d 4 + 35 d

1.105

−10 + 15 a −10 + 15 a

1.106

−56 + 105 y −56 + 105 y

1.107

− z + 11 − z + 11

1.108

− x + 4 − x + 4

1.109

3 − 3 x 3 − 3 x

1.110

2 x − 20 2 x − 20

1.111

5 x − 66 5 x − 66

1.112

7 x − 13 7 x − 13

Section 1.1 Exercises

1.

Divisible by 2, 3, 6

3.

Divisible by 2

5.

Divisible by 3, 5

7.

2 · 43 2 · 43

9.

5 · 7 · 13 5 · 7 · 13

11.

2 · 2 · 2 · 2 · 3 · 3 · 3 2 · 2 · 2 · 2 · 3 · 3 · 3

13.

24

15.

280

17.

440

19.

5

21.

58

23.

29

25.

149

27.

50

29.

5

31.

ⓐ 64 ⓑ 16 ⓒ 7

33.

21

35.

9

37.

10 x + 6 10 x + 6

39.

22 a + 1 22 a + 1

41.

17 x 2 + 20 x + 16 17 x 2 + 20 x + 16

43.

ⓐ 5x2−6xy5x2−6xy ⓑ 6y25x6y25x
ⓒ y2+21y2+21 ⓓ 81x2−6x81x2−6x

45.

ⓐ 4ab2+3a2b4ab2+3a2b ⓑ 20xy220xy2
ⓒ m+15m+15 ⓓ 121x2−9x121x2−9x

47.

ⓐ 8(y−9)8(y−9) ⓑ 8y−98y−9

49.

ⓐ 5(3x+y)5(3x+y) ⓑ 15x+y15x+y

51.

2 c + 14 2 c + 14

53.

3 n − 7 3 n − 7

55.

Answers will vary.

57.

Answers will vary.

Section 1.2 Exercises

59.

ⓐ >> ⓑ >> ⓒ >> ⓓ >>

61.

ⓐ == ⓑ == ⓒ >> ⓓ ==

63.

0

65.

8

67.

15

69.

1

71.

ⓐ −11−11 ⓑ −3−3 ⓒ 33

73.

32

75.

−22 −22

77.

29 29

79.

ⓐ 6 ⓑ −6−6 ⓒ −20−20 ⓓ 2020

81.

−59 −59

83.

22

85.

ⓐ 16 ⓑ 16

87.

ⓐ 45 ⓑ 45

89.

0

91.

4

93.

6

95.

ⓐ −32−32 ⓑ −65−65 ⓒ −4−4
ⓓ 1313

97.

ⓐ −4−4 ⓑ −12−12 ⓒ −39−39
ⓓ 1414

99.

ⓐ 6464 ⓑ −64−64

101.

−47 −47

103.

90 90

105.

9 9

107.

41 41

109.

−9 −9

111.

−29 −29

113.

1 1

115.

−3 −3

117.

5 5

119.

ⓐ −47−47 ⓑ 1616

121.

121 121

123.

−56 −56

125.

6 6

127.

( 3 + ( −15 ) ) + 7 ; − 5 ( 3 + ( −15 ) ) + 7 ; − 5

129.

ⓐ 10−(−18);2810−(−18);28
ⓑ −25−11;−36−25−11;−36

131.

−6 a + b −6 a + b

133.

96 ° 96 °

135.

21 yards

137.

− $ 28 − $ 28

139.

Answers will vary.

141.

Answers will vary.

Section 1.3 Exercises

143.

− 12 7 − 12 7

145.

10 21 10 21

147.

2 x 2 3 y 2 x 2 3 y

149.

− 21 a 2 11 b 2 − 21 a 2 11 b 2

151.

1 3 1 3

153.

− 21 50 − 21 50

155.

11 30 11 30

157.

9 n 9 n

159.

33 4 x 33 4 x

161.

− 4 9 − 4 9

163.

10 u 9 v 10 u 9 v

165.

− 1 16 − 1 16

167.

− 10 9 − 10 9

169.

− 2 5 − 2 5

171.

2 m 3 n 2 m 3 n

173.

29 24 29 24

175.

1 48 1 48

177.

17 105 17 105

179.

− 53 40 − 53 40

181.

1 12 1 12

183.

4 x + 3 12 4 x + 3 12

185.

ⓐ 5656 ⓑ 44

187.

ⓐ 25n1625n16 ⓑ 25n−163025n−1630

189.

ⓐ −8x−1518−8x−1518 ⓑ −10k27−10k27

191.

ⓐ −5(a+1)3−5(a+1)3 ⓑ aa

193.

9 7 9 7

195.

−8 −8

197.

11 6 11 6

199.

5 2 5 2

201.

54 54

203.

49 25 49 25

205.

15 4 15 4

207.

5 21 5 21

209.

5 4 5 4

211.

1 24 1 24

213.

−28 − 15 y 60 −28 − 15 y 60

215.

33 64 33 64

217.

7 9 7 9

219.

−5 −5

221.

23 24 23 24

223.

11 5 11 5

225.

1 1

227.

13 3 13 3

229.

ⓐ 1515 ⓑ 6565

231.

− 1 9 − 1 9

233.

− 5 11 − 5 11

235.

Answers will vary.

237.

Answers will vary.

Section 1.4 Exercises

239.

ⓐ 5.78 ⓑ 5.8 ⓒ 6

241.

ⓐ 0.30 ⓑ 0.3 ⓒ 0

243.

ⓐ 63.48 ⓑ 63.5 ⓒ 63

245.

−40.91 −40.91

247.

−7.22 −7.22

249.

−27.5 −27.5

251.

102.212 102.212

253.

51.31 51.31

255.

−4.89 −4.89

257.

−1200.47982 −1200.47982

259.

337.8914 337.8914

261.

1.305 1.305

263.

92.4 92.4

265.

2.5

267.

55200

269.

$ 2.44 $ 2.44

271.

−4.8 −4.8

273.

2.08 2.08

275.

1 25 1 25

277.

19 200 19 200

279.

0.85 0.85

281.

−12.4 −12.4

283.

0.71 0.71

285.

0.393 0.393

287.

156 % 156 %

289.

6.25 % 6.25 %

291.

8

293.

12

295.

−10 −10

297.

ⓐ 0,36,90,36,9 ⓑ −8,0,36,9−8,0,36,9 ⓒ −8,0,125,36,9−8,0,125,36,9 ⓓ 1.95286...,1.95286..., ⓔ −8,0,1.95286...,125,36,9−8,0,1.95286...,125,36,9

299.

ⓐ none ⓑ −100,−7,−1−100,−7,−1
ⓒ −100,−7,−83,−1,0.77,314−100,−7,−83,−1,0.77,314
ⓓ none
ⓔ −100,−7,−83,−1,0.77,314−100,−7,−83,−1,0.77,314

301.
Figure shows a number line with numbers ranging from 0 to 6. Some values are highlighted. From left to right, these are: 3 by 10, 11 by 6, 7 by 2 and 4.
303.
Figure shows a number line with numbers ranging from minus 4 to 4. Some values are highlighted. From left to right, these are: minus 5 by 2, minus 1 and two thirds, minus 3 by 4, 3 by 4, 1 and two thirds, and 5 by 2.
305.
Figure shows a number line with numbers ranging from minus 4 to 4. Two values are highlighted. One is between minus 2 and minus 1. The other is between 0 and 1.
307.
Figure shows a number line with numbers ranging from minus 4 to 4. Two values are highlighted. One is between minus 2 and minus 1. The other is between 3 and 4.
309.

Answers will vary.

311.

Answers will vary.

Section 1.5 Exercises

313.

27 m + ( −21 n ) 27 m + ( −21 n )

315.

5 4 g + 1 2 h 5 4 g + 1 2 h

317.

2.43 p + 8.26 q 2.43 p + 8.26 q

319.

−63 −63

321.

1 5 6 1 5 6

323.

17 17

325.

14.88 14.88

327.

10 p 10 p

329.

44 44

331.

7 8 7 8

333.

d d

335.

49 11 49 11

337.

0

339.

undefined

341.

undefined

343.

32 y + 72 32 y + 72

345.

6 c − 78 6 c − 78

347.

3 4 q + 3 3 4 q + 3

349.

5 y − 3 5 y − 3

351.

3 + 8 r 3 + 8 r

353.

36 d + 90 36 d + 90

355.

r s − 18 r r s − 18 r

357.

y p + 4 p y p + 4 p

359.

−28 p − 7 −28 p − 7

361.

−3 x + 18 −3 x + 18

363.

−3 x + 7 −3 x + 7

365.

−3 y − 8 −3 y − 8

367.

−33 c + 26 −33 c + 26

369.

− a + 19 − a + 19

371.

4 m − 10 4 m − 10

373.

72 x − 25 72 x − 25

375.

22 n + 9 22 n + 9

377.

6 c + 34 6 c + 34

379.

12 y + 63 12 y + 63

381.

Answers will vary.

383.

Answers will vary.

Review Exercises

385.

Divisible by 2,3,5,6,102,3,5,6,10

387.

120

389.

4

391.

3

393.

6 x 2 − x + 5 6 x 2 − x + 5

395.

ⓐ 11(y−2)11(y−2) ⓑ 11y−211y−2

397.

ⓐ == ⓑ == ⓒ >> ⓓ ==

399.

−9 −9

401.

ⓐ 8 ⓑ −8−8 ⓒ −22−22 ⓓ 22

403.

ⓐ 40 ⓑ 40

405.

ⓐ −3−3 ⓑ −15−15 ⓒ −56−56 ⓓ 17

407.

28

409.

−12 −12

411.

9

413.

( −4 + ( −9 ) ) + 23 ; 10 ( −4 + ( −9 ) ) + 23 ; 10

415.

46 ° 46 °

417.

− 15 x 3 11 y 2 − 15 x 3 11 y 2

419.

8 x 15 y 8 x 15 y

421.

31 36 31 36

423.

ⓐ 118118 ⓑ 5656

425.

− 1 6 − 1 6

427.

75

429.

− 1 5 − 1 5

431.

6.17 6.17

433.

96.978 96.978

435.

488.1813

437.

56.3 56.3

439.

−2.3 −2.3

441.

− 48 5 − 48 5

443.

1. 27 ¯ 1. 27 ¯

445.

4.75 % 4.75 %

447.

no real number

449.
Figure shows a number line with numbers ranging from minus 4 to 4. Some values are highlighted.
451.

3 4 x + y 3 4 x + y

453.

1 11 15 1 11 15

455.

9 17 9 17

457.

undefined

459.

8 b + 10 8 b + 10

461.

x p − 5 p x p − 5 p

463.

−6 x − 6 −6 x − 6

465.

6 y + 16 6 y + 16

Practice Test

467.

7 n + 7 7 n + 7

469.

−8 − 11 ; − 19 −8 − 11 ; − 19

( −8 − ( −3 ) ) + 5 ; 0 ( −8 − ( −3 ) ) + 5 ; 0

471.

ⓐ 28.1528.15 ⓑ 28.14628.146

473.
Figure shows a number line with numbers ranging from minus 4 to 4. Some values are highlighted.
475.

1

477.

−8 −8

479.

15 17 15 17

481.

− 5 3 − 5 3

483.

3 3

485.

− 7 6 − 7 6

487.

−65.4 −65.4

489.

−160

491.

1 8 13 1 8 13

493.

ⓐ 0 ⓑ undefined

495.

13 y − 3 13 y − 3

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