12.4.3: Chapter 3

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$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

Be Prepared

3.1

$.$

3.2

$<$

3.3

$14$

3.3

$20$

3.3

$3 + (- 7)$

3.4

$5$

3.5

$20 - (- 15)$

3.6

$11$

3.7

$20 \div 13$

3.8

$- 15$

3.9

$- 3$

3.10

$0$

3.11

$16$

3.12

$x - 5$

Try It

3.1 $This figure is a number line. The point negative 4 is labeled with the letter c, the point negative 1 is labeled with the letter b, and the point 1 is labeled with the letter a.$

3.2 $This figure is a number line. The point negative 4 is labeled with the letter a, the point negative 1 is labeled with the letter c, and the point 4 is labeled with the letter b.$

3.3

ⓐ >
ⓑ <
ⓒ >
ⓓ >

3.4

ⓐ <
ⓑ >
ⓒ <
ⓓ >

3.5

ⓐ −4
ⓑ 3

3.6

ⓐ −8
ⓑ 5

3.7

3.8

3.9

ⓐ −4
ⓑ 4

3.10

ⓐ −11
ⓑ 11

3.11

ⓐ 12
ⓑ −28

3.12

ⓐ 9
ⓑ −37

3.13

ⓐ 17
ⓑ 39
ⓒ −22
ⓓ −11

3.14

ⓐ 23
ⓑ 21
ⓒ −37
ⓓ −49

3.15

ⓐ >
ⓑ >
ⓒ <
ⓓ =

3.16

ⓐ>
ⓑ =
ⓒ >
ⓓ <

3.17

ⓐ 3
ⓑ 18

3.18

ⓐ 11
ⓑ 63

3.19

3.20

3.21

3.22

3.23

ⓐ −9
ⓑ 15
ⓒ −20
ⓓ 11−(−4)

3.24

ⓐ 19
ⓑ −22
ⓒ −9
ⓓ −8−(−5)

3.25

5 yards

3.26

−30 feet

3.27 $This figure has six pink circles in a row, representing positive counters. The first two circles are separated from the following four circles.$

3.28 $This figure has seven pink circles in a row, representing positive counters. Two circles are separated from the following 5 circles.$

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

3.34 $This figure shows two rows of counters. The first row shows 2 dark pink circles, representing negative counters. The second row has 5 light pink circles, representing negative counters. The neutral pairs of one positive and one negative counter are circled leaving three positive counters.$

3.35

ⓐ
$.$
ⓑ
$.$
ⓒ
$.$
ⓓ
$.$

3.36

ⓐ
$.$
ⓑ
$.$
ⓒ
$.$
ⓓ
$.$

3.37

ⓐ −17
ⓑ 57

3.38

ⓐ −46
ⓑ 26

3.39

−50

3.40

−70

3.41

3.42

3.43

ⓐ 2
ⓑ −12

3.44

ⓐ 2
ⓑ −1

3.45

ⓐ −6
ⓑ 10

3.46

ⓐ −1
ⓑ 17

3.47

3.48

3.49

196

3.50

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

3.60 $This figure shows a row of 7 light pink circles representing positive counters. The first four counters are circled.$

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

3.66 $This figure shows a row of 11 light pink circles, representing positive counters. The first seven counters are separated from the last four. Below the light pink circles are four dark pink circles, representing negative counters. These four negative counters are circled.$

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
ⓑ

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

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
ⓑ

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

3.71

ⓐ 8, 8
ⓑ −18, −18

3.72

ⓐ 8, 8
ⓑ −22, −22

3.73

ⓐ 19, 19
ⓑ −4, −4

3.74

ⓐ 23, 23
ⓑ 3, 3

3.75

−29

3.76

−26

3.77

3.78

3.79

−69

3.80

−47

3.81

ⓐ −2
ⓑ −15

3.82

ⓐ −2
ⓑ −7

3.83

ⓐ −2
ⓑ 36

3.84

ⓐ −19
ⓑ 9

3.85

ⓐ 14 − (−23) = 37
ⓑ −17 − 21 = −38

3.86

ⓐ 11 − (−19) = 30
ⓑ −11 − 18 = −29

3.87

45 degrees Fahrenheit

3.88

9 degrees Fahrenheit

3.89

10,103 feet

3.90

233 feet

3.91

ⓐ $48
ⓑ −$2
ⓒ $18

3.92

ⓐ −$54
ⓑ No, −$5

3.93

ⓐ −48
ⓑ 28
ⓒ −63
ⓓ 60

3.94

ⓐ −56
ⓑ 54
ⓒ −28
ⓓ 39

3.95

ⓐ −9
ⓑ 17

3.96

ⓐ −8
ⓑ 16

3.97

ⓐ −7
ⓑ 39

3.98

ⓐ −9
ⓑ 23

3.99

ⓐ −6
ⓑ 36

3.100

ⓐ −28
ⓑ 52

3.101

−63

3.102

−84

3.103

ⓐ 81
ⓑ −81

3.104

ⓐ 49
ⓑ −49

3.105

3.106

3.107

3.108

3.109

3.110

3.111

3.112

3.113

−8

3.114

3.115

−5 (12) = −60

3.116

8 (−13) = −104

3.117

−63 ÷ −9 = 7

3.118

−72 ÷ −9 = 8

3.119

ⓐ no
ⓑ no
ⓒ yes

3.120

ⓐ no
ⓑ no
ⓒ 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

3.127

3.128

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. $This figure is a number line. Negative 8 is labeled a, negative 6 is labeled c, and 5 is labeled b.$

ⓐ >
ⓑ <
ⓒ <
ⓓ >

ⓐ <
ⓑ >
ⓒ <
ⓓ >

ⓐ −2
ⓑ 6

11.

ⓐ 8
ⓑ −1

13.

15.

17.

ⓐ −3
ⓑ 3

19.

ⓐ −12;
ⓑ 12

21.

ⓐ 7
ⓑ 25
ⓒ 0

23.

ⓐ 32
ⓑ 18
ⓒ 16

25.

ⓐ 28
ⓑ 15

27.

ⓐ −19
ⓑ −33

29.

ⓐ <
ⓑ =

31.

ⓐ >
ⓑ >

33.

35.

37.

39.

41.

43.

ⓐ −8
ⓑ −(−6), or 6
ⓒ −3
ⓓ 4−(−3)

45.

ⓐ −20
ⓑ −(−5), or 5
ⓒ −12
ⓓ 18−(−7)

47.

−6 degrees

49.

−40 feet

51.

−12 yards

53.

55.

57.

ⓐ 20,320 feet
ⓑ −282 feet

59.

ⓐ $540 million
ⓑ −$27 billion

61.

Sample answer: I have experienced negative temperatures.

Section 3.2 Exercises

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

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

71.

−80

73.

75.

−135

77.

79.

−22

81.

108

83.

−4

85.

87.

ⓐ −18
ⓑ −87

89.

ⓐ −47
ⓑ 16

91.

ⓐ −4
ⓑ 10

93.

ⓐ −13
ⓑ 5

95.

−8

97.

99.

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 yards

119.

25-yard line

121.

20 feet

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

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

135.

ⓐ 9
ⓑ 9

137.

ⓐ 16
ⓑ 16

139.

ⓐ 17
ⓑ 17

141.

ⓐ 45
ⓑ 45

143.

145.

147.

−39

149.

−48

151.

−42

153.

−59

155.

−51

157.

159.

−2

161.

−2

163.

165.

167.

−20

169.

171.

173.

175.

–8

177.

−11

179.

ⓐ −3
ⓑ −9

181.

ⓐ 3
ⓑ 7

183.

−8

185.

−192

187.

ⓐ 3 − (−10) = 13
ⓑ 45 − (−20) = 65

189.

ⓐ −6 − 9 = −15
ⓑ −16 − (−12) = −4

191.

ⓐ −17 − 8 = −25
ⓑ −24 − 37 = −61

193.

ⓐ 6 − 21 = −15
ⓑ −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.

217.

−63

219.

−6

221.

223.

−4

225.

−8

227.

229.

−12

231.

−49

233.

−47

235.

237.

−125

239.

241.

−16

243.

245.

−88

247.

249.

251.

−5

253.

−9

255.

−29

257.

259.

ⓐ 1
ⓑ 33

261.

ⓐ −5
ⓑ 25

263.

265.

267.

269.

−56

271.

−3·15 = −45

273.

−60 ÷ (−20) = 3

275.

$\frac{−6}{a + b}$

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 the first expression the base is positive and after you raise it to the power you should take the opposite. Then in the second expression the base is negative so you simply raise it to the power.

Section 3.5 Exercises

285.

ⓐ no
ⓑ no
ⓒ yes

287.

ⓐ no
ⓑ no
ⓒ yes

289.

n = −7

291.

p = −17

293.

u = −4

295.

h = 6

297.

x = −16

299.

r = −14

301.

3x = 6; x = 2

303.

2x = 8; x = 4

305.

x = 9

307.

c = −8

309.

p = 3

311.

q = −12

313.

x = 20

315.

z = 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.

−15f = 75; f = −5

327.

−6 + c = 4; c = 10

329.

m − 9 = −4; m = 5

331.

ⓐ x = 8
ⓑ x = 5

333.

ⓐ p = −9
ⓑ p = 30

335.

a = 20

337.

m = 7

339.

u = −52

341.

r = −9

343.

d = 5

345.

x = −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.

369.

ⓐ −8
ⓑ 8

371.

ⓐ −32
ⓑ 32

373.

375.

377.

379.

381.

−33

383.

385.

387.

55; −55

389.

391.

393.

−1

395.

−16

397.

−3

399.

−10°

401.

403.

405.

407.

−50

409.

−1

411.

413.

ⓐ 3
ⓑ −16

415.

−27

417.

−8 + 2 = −6

419.

10 + [−5 + (−6)] = −1

421.

16 degrees

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

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

427.

429.

−38

431.

−58

433.

−1

435.

ⓐ −2
ⓑ −11

437.

439.

−12 − 5 = −17

441.

−2 degrees

443.

−36

445.

121

447.

−7

449.

−8

451.

−45

453.

−9

455.

−81

457.

459.

461.

−66

463.

−58

465.

−12(6) = −72

467.

ⓐ no
ⓑ yes
ⓒ no

469.

−12

471.

−7

473.

3x = 9; x = 3

475.

477.

479.

−6y = −42; y = 7

481.

m + 4 = −48; m = −52

483.

Answers will vary.

Practice Test

485.

ⓐ <
ⓑ >

487.

ⓐ 7
ⓑ −8

489.

491.

−27

493.

495.

497.

−8

499.

501.

503.

505.

−7 − (−4) = −3

507.

4°F

509.

n = −1

511.

r = 6

513.

y − 8 = −32; y = −24

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