Chapter 2 - Intermediate Algebra

Be Prepared

2.1

$18 x + 30$

2.2

$3 - 2 n$

2.3

$12$

2.4

$2 x - 6$

2.5

$0.045$

2.6

$60 %$

2.7

$16$

2.8

$w - 3$

2.9

$63$

2.10

$0.35 x + 0.4$

2.11

$2 c + 3$

2.12

$0.042$

2.13

$15$ is greater than $x$ .

2.14

$15 < x$

2.15

$\frac{2}{5} x + 4$

2.16

$- x + 4$

2.17

$- 7$

2.18

a. >; b. >; c. >; d. >

2.19

$12$

Try It

2.1

ⓐ no ⓑ yes

2.2

ⓐ yes ⓑ no

2.3

$m = 2$

2.4

$a = 0$

2.5

$u = 2$

2.6

$x = 4$

2.7

$p = −2$

2.8

$q = −8$

2.9

$y = - \frac{17}{5}$

2.10

$z = 0$

2.11

identity; all real numbers

2.12

identity; all real numbers

2.13

conditional equation; $q = - \frac{9}{11}$

2.14

conditional equation; $k = \frac{201}{14}$

2.15

contradiction; no solution

2.16

contradiction; no solution

2.17

$x = \frac{1}{2}$

2.18

$x = −2$

2.19

$x = 12$

2.20

$u = −12$

2.21

$n = 2$

2.22

$m = −1$

2.23

$r = 1$

2.24

$s = −8$

2.25

$n = 9$

2.26

$d = 16$

2.27

He bought two notebooks.

2.28

He did seven crosswords puzzles.

2.29

$3$

2.30

$6$

2.31

$−15, −8$

2.32

$−29, 11$

2.33

$−33, −32, −31$

2.34

$−13, −12, −11$

2.35

32, 34, 36

2.36

$−10, −8, −6$

2.37

The average cost was $5,000.

2.38

The median price was $19,300.

2.39

ⓐ 36 ⓑ $26 ⓒ $125 %$

2.40

ⓐ 33 ⓑ $36 ⓒ $175 %$

2.41

25 grams

2.42

2,375 mg

2.43

50%

2.44

40%

2.45

$8.8 %$

2.46

50%

2.47

ⓐ $600 ⓑ $1,800

2.48

ⓐ $2,975 ⓑ $11,475

2.49

He will earn $2,500.

2.50

She earned $7,020.

2.51

The rate of simple interest was 6%.

2.52

The rate of simple interest was 5.5%.

2.53

He paid $17,590.

2.54

She deposited $9,600.

2.55

$b = \frac{2 A}{h}$

2.56

$h = \frac{2 A}{b}$

2.57

$C = \frac{5}{9} (F - 32)$

2.58

$b = \frac{2 A - B h}{h}$

2.59

$t = \frac{A - P}{P r}$

2.60

$r = \frac{A - P}{P t}$

2.61

$y = \frac{9 - 4 x}{7}$

2.62

$y = \frac{1 - 5 x}{8}$

2.63

The window’s height is 12 meters.

2.64

The length of the base is 6 feet.

2.65

The measures of the angles are 20°, 70°, and 90°.

2.66

The measures of the angles are 30°, 60°, and 90°.

2.67

The length of the leg is 8.

2.68

The length of the leg is 12.

2.69

The length is 39 inches and the width is 16 inches.

2.70

The length is 17 yards and the width is 26 yards.

2.71

The lengths of the sides of the triangle are 5, 11 and 12 inches.

2.72

The lengths of the sides of the triangle are 4, 7 and 9 feet.

2.73

The length of the swimming pool is 70 feet and the width is 30 feet.

2.74

The length of the garden is 90 yards and the width is 60 yards.

2.75

The ladder reaches 12 feet.

2.76

He should attach the lights 8 feet from the base of the mast.

2.77

Jess has 41 nickels and 18 quarters.

2.78

Elane has 22 nickels and 59 dimes.

2.79

Eric bought thirty-two 49-cent stamps and twelve 35-cent stamps.

2.80

Kailee bought twenty-six 49-cent stamps and ten 20-cent stamps.

2.81

84 adult tickets, 31 student tickets

2.82

615 children’s tickets and 195 adult tickets

2.83

Orlando mixed five pounds of cereal squares and 25 pounds of nuts.

2.84

Becca mixed 21 gallons of fruit punch and seven gallons of soda.

2.85

The speed of the local train is 48 mph and the speed of the express train is 60 mph.

2.86

Jeromy drove at a speed of 80 mph and his mother drove 60 mph.

2.87

Christopher’s speed was 50 mph and his parents’ speed was 40 mph.

2.88

Ashley’s parents drove 55 mph and Ashley drove 62 mph.

2.89

Pierre and Monique will be 429 miles apart in 3 hours.

2.90

Thanh and Nhat will be 330 miles apart in 2.2 hours.

2.91

Suzy’s speed uphill is $1.8$ mph and downhill is three mph.

2.92

The boat’s speed upstream is eight mph and downstream is12 mph.

2.93

Hamilton drove 40 mph in the city and 70 mph in the desert.

2.94

Phuong rode uphill at a speed of 12 mph and on the flat street at 20 mph.

2.95

ⓐ

ⓑ

The graph of the inequality x is less than or equal to negative 1.5 is indicated on a number line with a right bracket at negative 1.5 and shading to the left. The solution in interval notation is the interval from negative infinity to negative 1.5 enclosed within a left parenthesis and right bracket.

ⓒ

The graph of the inequality x is greater than or rqual to three-fourths is indicated on a number line with a left bracket at three-fourths and shading to the right. The solution in interval notation is the interval from three-fourths to infinity enclosed within a left bracket and left parentheses.

2.96

ⓐ

ⓑ

ⓒ

The graph of the inequality x is less than negative two-thirds is indicated on a number line with a right parenthesis at negative two-thirds and shading to the left. The solution in interval notation is the interval from negative infinity to negative two-thirds enclosed within parentheses.

2.97

ⓐ

Negative 2 is less x which is less than 1. There are open circles at negative 2 and 1 and shading between negative 2 and 1 on the number line. Put parentheses at negative 2 and 1. Write in interval notation.

ⓑ

Negative 5 is less than or equal to x which is less than negative 4. There is a closed circle at negative 6 and an open circle at negative 4 and shading between negative 5 and negative 4 on the number line. Put a bracket at negative 5 and a parenthesis at negative 4. Write in interval notation.

ⓒ

1 is less than or equal to x which is less than 4.25. There is closed circle at 1 and a closed circle at 4.25 and shading between 1 and 4.25 on the number line. Put brackets at 1 and 4.25. Write in interval notation.

2.98

ⓐ

Negative 6 is less than x which is less than 2. There is an open circle at negative 6 and an open circle at 2 and shading between negative 6 and 2 on the number line. Put parentheses at negative 6 and 2. Write in interval notation.

ⓑ

Negative 3 is less than or equal to x which is less than negative 1. There is a closed circle at negative 3 and an open circle at negative 1 and shading between negative 3 and negative 1 on the number line. Put a bracket at negative 3 and a parenthesis at negative 1. Write in interval notation.

ⓒ

2.5 is less than or equal to x which is less thanor equal to 6. There is a closed circle at 2.5 and a closed circle at 6 and shading between 2.5 and 6 on the number line. Put brackets at 2.5 and 6. Write in interval notation.

2.99

ⓐ

p is less than eleven-twelfths. The solution on the number line has a right bracket at eleven-twelfths with shading to the right. The solution in interval notation is, eleven-twelfths to infinity within a bracket and parenthesis.

ⓑ

c is less than 8. The solution on the number line has a left bracket at 8 with shading to the right. The solution in interval notation is, 8 to infinity within parentheses.

ⓒ

m is greater than or equal to 8. The solution on the number line has a right bracket at 64 with shading to the right. The solution in interval notation is, 64 to infinity within a bracket and parentheses.

2.100

ⓐ

r is less than or equal to eleven-twelfths. The solution on the number line has a left bracket at eleven-twelfths with shading to the left. The solution in interval notation is negative infinity to eleven-twelfths within a parenthesis and a bracket.

ⓑ

c is less than or equal to 5. The solution on the number line has a right bracket at 5 with shading to the left. The solution in interval notation is negative infinity to 5 within a parentheses and a bracket.

ⓒ

n is greater than negative 18. The solution on the number line has a left parenthesis at negative 18 with shading to the right. The solution in interval notation is negative 18 to infinity within parentheses.

2.101

ⓐ

q is greater than or equal to negative 4. The solution on the number line has a left parenthesis at negative 4 with shading to the right. The solution in interval notation is negative 4 to infinity within parentheses.

ⓑ

k is greater than or equal to negative 180. The solution on the number line has a left bracket at negative 180 with shading to the right. The solution in interval notation is negative 180 to infinity within a bracket and a parenthesis.

2.102

ⓐ

r is greater than or equal to 10. The solution on the number line has a left bracket at 10 with shading to the right. The solution in interval notation is 10 to infinity within a bracket and parenthesis.

ⓑ

u is less than or equal to 64. The solution on the number line has a right bracket at 64 with shading to the left. The solution in interval notation is negative infinity to 64 within parenthesis and a bracket.

2.103

q is less than or equal to 23 divided by 4. The solution on the number line has a right bracket at 23 divided by 4 with shading to the left. The solution in interval notation is negative infinity to 23 divided by 4 within a parenthesis and a bracket.

2.104

x is greater than negative 19 divided by 4. The solution on the number line has a left parenthesis at negative 19 divided by 4 with shading to the right. The solution in interval notation is negative 19 divided by 4 to infinity within parentheses.

2.105

y is greater than negative 6. The solution on the number line has a left parenthesis at negative 6 with shading to the right. The solution in interval notation is negative 6 to infinity within parentheses.

2.106

u is greater than negative 10. The solution on the number line has a left parenthesis at 10 with shading to the right. The solution in interval notation is 10 to infinity within parentheses.

2.107

The inequality is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.

2.108

2.109

The inequality is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation.

2.110

2.111

p minus 19 is greater than or equal to 47. Its solution is p is greater than or equal to 66. The solution on the number line has a left bracket at 66 with shading to the right. The solution in interval notation is 66 to infinity within a bracket and a parenthesis.

2.112

a plus 4 is less than or equal to 15. Its solution is a is less than or equal to 11. The solution on the number line has a right bracket at 11with shading to the left. The solution in interval notation is negative infinity to 11 within a parenthesis and bracket.

2.113

Angie can buy 7 packs of juice.

2.114

Daniel can have 11 people at the party.

2.115

Sergio and Lizeth can travel no more than 500 miles.

2.116

Rameen can use no more than 76 therms.

2.117

Caleb must work at least 96 hours.

2.118

Elliot must work at least 85 jobs.

2.119

Brenda must babysit at least 27 hours.

2.120

Josue must shovel at least 20 driveways.

2.121

The solution is negative 1 is less than or equal to x which is less than 4. On a number line it is shown with a closed circle at negative 1 and an open circle at 4 with shading in between the closed and open circles. Its interval notation is negative 1 to 4 within a bracket and a parenthesis.

2.122

The solution is negative 2 is less than or equal to x which is less than 3. On a number line it is shown with a closed circle at negative 2 and an open circle at 3 with shading in between the closed and open circles. Its interval notation is negative 2 to 3 within a bracket and a parenthesis.

2.123

The solution is x is less than three-halves. On a number line it is shown with an open circle at three-halves with shading to its left. Its interval notation is negative infinity to three-halves within a parentheses.

2.124

The solution is x is less than negative 1. On a number line it is shown with an open circle at 1 with shading to its left. Its interval notation is negative infinity to negative 1 within parentheses.

2.125

The inequality is a contradiction. So, there is no solution. As a result, there is no graph of the number line or interval notation.

2.126

The inequality is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.

2.127

The solution is negative 1 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 1 and an open circle at 2 with shading between the closed and open circles. Its interval notation is negative 1 to 2 within a bracket and a parenthesis.

2.128

The solution is 1 is less than x which is less than or equal to 3. Its graph has an open circle at 1 and a closed circle at 3 with shading between the closed and open circles. Its interval notation is negative 1 to 3 within a parenthesis and a bracket.

2.129

The solution is x is greater than or equal to 2 or x is less than or equal to 1. The graph of the solutions on a number line has a closed circle at negative 1 and shading to the left and a closed circle at 2 with shading to the right. The interval notation is the union of negative infinity to negative 1 within a parenthesis and a bracket and 2 and infinity within a bracket and a parenthesis.

2.130

The solution is x is greater than or equal to 1 or x is less than or equal to negative 1. The graph of the solutions on a number line has a closed circle at negative 1 and shading to the left and a closed circle at 1 with shading to the right. The interval notation is the union of negative infinity to negative 1 within a parenthesis and a bracket and 1 and infinity within a bracket and a parenthesis.

2.131

The solution is an identity. Its solution on the number line is shaded for all values. The solution in interval notation is negative infinity to infinity within parentheses.

2.132

2.133

The homeowner can use 5–20 hcf and still fall within the “conservation usage” billing range.

2.134

The homeowner can use 16–40 hcf and still fall within the “normal usage” billing range.

2.135

ⓐ $± 2$ ⓑ no solution ⓒ 0

2.136

ⓐ $± 11$ ⓑ no solution ⓒ 0

2.137

$x = 4, x = - \frac{2}{3}$

2.138

$x = −1, x = \frac{5}{2}$

2.139

$x = 8, x = 0$

2.140

$x = 8, x = 2$

2.141

No solution

2.142

No solution

2.143

$x = - \frac{2}{5},$ $x = \frac{5}{2}$

2.144

$x = 3,$ $x = \frac{1}{9}$

2.145

The solution is negative 9 is less than x which is less than 9. The number line shows open circles at negative 9 and 9 with shading in between the circles. The interval notation is negative 9 to 9 within parentheses.

2.146

The solution is negative 1 is less than x which is less than 1. The number line shows open circles at negative 1 and 1 with shading in between the circles. The interval notation is negative 1 to 1 within parentheses.

2.147

The solution is negative 2 is less than or equal to x which is less than or equal to 3. The number line shows closed circles at negative 2 and 3 with shading between the circles. The interval notation is negative 2 to 3 within brackets.

2.148

The solution is one-half is less than or equal to x which is less than or equal to 2. The number line shows closed circles at one-half and 2 with shading between the circles. The interval notation is one-half to 2 within brackets.

2.149

The solution is x is less than negative 2 or x is greater than 2. The number line shows an open circle at negative 2 with shading to its left and an open circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative 2 within parentheses and 2 to infinity within parentheses.

2.150

The solution is x is less than negative 1 or x is greater than 1. The number line shows an open circle at negative 1 with shading to its left and an open circle at 1 with shading to its right. The interval notation is the union of negative infinity to negative 1 within parentheses and 1 to infinity within parentheses.

2.151

The solution is x is less than or equal to negative one-half or x is greater than or equal 2. The number line shows a closed circle at negative one-half with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative one-half within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis

2.152

The solution is x is less than or equal to two-thirds or x is greater than or equal 2. The number line shows a closed circle at two-thirds with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to two-thirds within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.

2.153

The diameter of the rod can be between 79.991 and 80.009 mm.

2.154

The diameter of the rod can be between 74.95 and 75.05 mm.

Section 2.1 Exercises

1.

ⓐ yes ⓑ no

3.

ⓐ no ⓑ yes

5.

$y = 5$

7.

$w = −18$

9.

$q = 3$

11.

$x = 14$

13.

$c = 4$

15.

$c = \frac{5}{2}$

17.

$n = 2$

19.

$y = −5$

21.

$s = 10$

23.

$p = −4$

25.

$c = −4$

27.

$h = \frac{3}{4}$

29.

$m = 6$

31.

identity; all real numbers

33.

conditional equation; $j = \frac{2}{5}$

35.

conditional equation; $m = \frac{16}{5}$

37.

contradiction; no solution

39.

contradiction; no solution

41.

identity; all real numbers

43.

$x = −1$

45.

$y = −1$

47.

$a = \frac{3}{4}$

49.

$x = 4$

51.

$w = \frac{9}{4}$

53.

$b = 12$

55.

$p = −41$

57.

$x = - \frac{5}{2}$

59.

$n = −3$

61.

$x = −2$

63.

$u = 3$

65.

$x = 18$

67.

$x = 20$

69.

$n = 9$

71.

$d = 8$

73.

$L = 19.75$ feet

75.

Answers will vary.

77.

Answers will vary.

79.

Answers will vary.

Section 2.2 Exercises

81.

Answers will vary.

83.

six boys

85.

58 hardback books

87.

15

89.

5

91.

12

93.

$−5$

95.

18, 24

97.

8, 12

99.

32, 46

101.

4, 10

103.

38, 39

105.

25, 26, 27

107.

$−11, −12, −13$

109.

84, 86, 88

111.

$−69, −71, −73$

113.

$750

115.

$13,500

117.

ⓐ 54 ⓑ 108 ⓐ 30%

119.

ⓐ $162.5$ ⓑ $35 ⓐ 150%

121.

$$ 11.88$

123.

24.2 g

125.

2407 mg

127.

45%

129.

25%

131.

12%

133.

200%

135.

$−2.5 %$

137.

$−11 %$

139.

ⓐ $$ 26.97$ ⓑ $$ 17.98$

141.

ⓐ $576 ⓑ 30%

143.

ⓐ $$ 7.20$ ⓑ $$ 23.20$

145.

ⓐ $$ 0.20$ ⓑ $$ 0.80$

147.

$116

149.

$4836

151.

3%

153.

$3.75 %$

155.

$35,000

157.

$3345

159.

$21.2 %$

161.

Answers will vary.

163.

Answers will vary.

Section 2.3 Exercises

165.

$d = \frac{C}{π}$

167.

$L = \frac{V}{W H}$

169.

$b = \frac{2 A}{h}$

171.

$d_{1} = \frac{2 A}{d_{2}}$

173.

$b_{1} = \frac{2 A}{h} - b_{2}$

175.

$a = \frac{2 h - 108 t}{t^{2}}$

177.

$a = 180 - b - c$

179.

$p = \frac{2 A - 2 B}{l}$

181.

$L = \frac{P - 2 W}{2}$

183.

$y = 15 - 8 x$

185.

$y = −6 + 4 x$

187.

$y = 4 + x$

189.

$y = \frac{7 - 4 x}{3}$

191.

$y = \frac{12 - 2 x}{3}$

193.

$y = \frac{18 - 3 x}{−2}$

195.

1 foot

197.

23 inches

199.

$45 °, 45 °, 90 °$

201.

$30 °, 60 °, 90 °$

203.

15

205.

25

207.

8

209.

12

211.

$10.2$

213.

$9.8$

215.

18 meters, 11 meters

217.

$13.5$ m, $12.8$ m

219.

25 ft, 50 ft

221.

l = 11 m, w = 7 m

223.

12 ft, 13 ft, 14 ft

225.

3 ft, 6 ft, 8 ft

227.

120 yd, 160 yd

229.

40 ft, 85 ft

231.

5 feet

233.

14.1 feet

235.

$104 ° F$

237.

240 ft

239.

Answers will vary.

241.

ⓐ Answers will vary. ⓑ The areas are the same. The $2 \times 8$ rectangle has a larger perimeter than the $4 \times 4$ square.
ⓒ Answers will vary.

Section 2.4 Exercises

243.

nine nickels, 16 dimes

245.

ten $10 bills, five $5 bills

247.

63 dimes, 20 quarters

249.

16 nickels, 12 dimes, seven quarters

251.

330 day passes, 367 tournament passes

253.

40 postcards, 100 stamps

255.

15 $10 shares, five $12 shares

257.

34 general, 61 youth

259.

114 general, 246 student

261.

Four pounds of macadamia nuts, eight pounds almonds

263.

$3.6$ lbs Bermuda seed, $5.4$ lbs Fescue seed

265.

$33,000 in Fund A, $22,000 in Fund B

267.

$5.9 %$

269.

Kathy 5 mph, Cheryl 3 mph

271.

commercial 550 mph, private plane 340 mph

273.

Violet 65 mph, Charlie 55 mph

275.

Ethan 22 mph, Leo 16 mph

277.

DaMarcus 16 mph, Fabian 22 mph

279.

four hours

281.

4.5 hours

283.

uphill $1.6$ mph, downhill $4.8$ mph

285.

light traffic 54 mph, heavy traffic 30 mph

287.

freeway 67 mph, mountain road 22.3 mph

289.

running eight mph, walking three mph

291.

ⓐ 15 minutes ⓑ 20 minutes ⓒ one hour (d) 1:25

293.

Answers will vary.

295.

Answers will vary.

Section 2.5 Exercises

297.

ⓐ

ⓑ

ⓒ

The solution for x is greater than or equal to one-third on a number line has a left bracket at one-third with shading to the right. The solution in interval notation is one-third to infinity within a bracket and a parenthesis.

299.

ⓐ

ⓑ

ⓒ

The solution for x is less than negative seven-thirds on a number line has a right parenthesis with shading to the left. The solution in interval notation is negative infinity to negative seven-thirds within parentheses.

301.

ⓐ

Negative 2 is less than x which is less than 0. There is an open circle at negative 2 and an open circle at 0 and shading between negative 2 and 0 on the number line. The interval notation is negative 2 and 0 within parentheses.

ⓑ

Negative 5 is less than or equal to x which is less than negative 3. There is a closed circle at negative 5 and an open circle at negative 3 and shading between negative 5 and negative 3 on the number line. The interval notation is negative 5 and negative 3 within a bracket and a parenthesis.

ⓒ

0 is less than or equal to x which is less than or equal to 3.5. There is a closed circle at 0 and a closed circle at 3.5 and shading between 0 and 3.5 on the number line. The interval notation is 0 and 3.5 within brackets.

303.

ⓐ

Negative 4 is less than x which is less than 2. There is a open circle at negative 4 and an open circle at 2 and shading between negative 4 and 2 on the number line. The interval notation is negative 4 and 2 within parentheses.

ⓑ

Negative 5 is less than x which is less than or equal to 2. There is a open circle at negative 5 and a closed circle at negative 2 and shading between negative 5 and negative 2 on the number line. The interval notation is negative 5 and negative 2 within a parenthesis and a bracket.

ⓒ

Negative 3.75 is less than or equal to x which is less than or equal to 0. There is a closed circle at negative 3.75 and a closed circle at 0 and shading between negative 3.75 and 0 on the number line. The interval notation is negative 3.75 and 0 within brackets.

305.

ⓐ

The solution is b is greater than or equal to negative seventeen twenty-fourths. The solution on a number line has a left bracket negative seventeen twenty-fourths with shading to the right. The solution in interval notation is negative seventeen twenty-fourths to infinity within a bracket and a parenthesis.

ⓑ

The solution is y is less than 8. The solution on a number line has a right parenthesis at 8 with shading to the left. The solution in interval notation is negative infinity to 8 within parentheses.

ⓒ

The solution is k is greater than 64. The solution on a number line has a left parenthesis at 64 with shading to the right. The solution in interval notation is 64 to infinity within parentheses.

307.

ⓐ

The solution is g is less than twenty-three thirty-sixths. The solution on a number line has a right parenthesis at twenty-three thirty-sixths with shading to the left. The solution in interval notation is negative infinity to twenty-three thirty-sixths within parentheses.

ⓑ

The solution is s is less than negative 4. The solution on a number line has a right parenthesis at negative 4 with shading to the left. The solution in interval notation is negative infinity to negative 4 within parentheses.

ⓒ

The solution is g is less than or equal to 16. The solution on a number line has a right bracket at 16 with shading to the left. The solution in interval notation is negative infinity to 16 within parenthesis and a bracket.

309.

ⓐ

The solution is v is greater than or equal to negative 12. The solution on a number line has a left bracket with shading to the right. The solution in interval notation is negative 12 to infinity within a bracket and a parenthesis.

ⓑ

The solution is b is less than or equal to negative 300. The solution on a number line has a right bracket at negative 300 with shading to the left. The solution in interval notation is negative infinity to negative 300 within a parenthesis and a bracket.

311.

ⓐ

The solution is d is less than negative 15. The solution on a number line has a right parentheses with shading to the left. The solution in interval notation is negative infinity to negative 15 within parentheses.

ⓑ

The solution is q is greater than 108. The solution on a number line has a left parentheses at 108 with shading to the right. The solution in interval notation is 108 to infinity within parentheses.

313.

The solution is u is greater than or equal to 7. The solution on a number line has a left bracket at 7 with shading to the right. The solution in interval notation is 7 to infinity within a a bracket and a parenthesis.

315.

The solution is p is less than eighteen fifths. The solution on a number line has a right parenthesis at eighteen fifths with shading to the left. The solution in interval notation negative infinity to eighteen fifths within parentheses.

317.

The solution is y is less than negative 5. The solution on a number line has a right parenthesis at negative 5 with shading to the left. The solution in interval notation is negative infinity to negative 5 within parentheses.

319.

The solution is k is less than or equal to 7. The solution on a number line has a right bracket at 7with shading to the left. The solution in interval notation is negative infinity to 7 within a parenthesis and a bracket.

321.

323.

325.

327.

329.

The solution is k is greater than or equal to negative eleven fifthss. The solution on a number line has a left bracket at negative eleven fifths with shading to the right. The solution in interval notation is negative eleven fifths to negative infinity within a bracket and a parenthesis.

331.

333.

The solution is y is greater than or equal to ten twenty-ninths. The solution on a number line has a left bracket at ten twenty-ninths with shading to the right. The solution in interval notation is ten twenty-ninths to infinity within a bracket and a parenthesis.

335.

The solution is g is greater than negative 90. The solution on a number line has a left parenthesis at negative 90 with shading to the right. The solution in interval notation is negative 90 to infinity within parentheses.

337.

The solution is n is less than or equal to negative 78. The solution on a number line has a right bracket at negative 78 with shading to the left. The solution in interval notation is negative infinity to negative 78 within a parenthesis and a bracket.

339.

The inequality is k plus 6 is greater than 25. Its solution is k is greater than 19. The solution on a number line has a left parenthesis at 19 with shading to the right. The solution in interval notation is 19 to infinity within parentheses.

341.

The inequality is x minus 12 is greater than or equal to 21. Its solution is x is greater than or equal to 33. The solution on a number line has a left bracket at 33 with shading to the right. The solution in interval notation is 33 to infinity within a bracket and a parenthesis.

343.

The inequality is negative 2 s is less than 56. Its solution is s is greater than negative 28. The solution on a number line has a left parenthesis at negative 28 with shading to the right. The solution in interval notation is negative 28 to infinity within parentheses.

345.

The inequality is a minus 15 is greater than or equal to negative 7. Its solution is a is greater than or equal to 8. The solution on a number line has a left bracket at 8 with shading to the right. The solution in interval notation is 8 to infinity within a bracket and a parenthesis.

347.

A maximum of 14 people can safely ride in the elevator.

349.

five drinks

351.

86 cars

353.

$16,875

355.

$110,000

357.

260 messages

359.

35 people

361.

32 jobs

363.

62 necklaces

365.

seven lawns

367.

20 hours

369.

20 people

371.

42 guests

373.

Answers will vary.

375.

Answers will vary.

Section 2.6 Exercises

377.

The solution is negative 2 is less than x which is less than or equal to 4. Its graph has an open circle at 1negative 2 and a closed circle at 4 with shading between the open and closed circles. Its interval notation is negative 2 to 4 within a parenthesis and a bracket.

379.

The solution is negative 6 is less than x which is less than negative 3. Its graph has an open circle at negative 6 and an open circle at negative 3 with shading between open circles. Its interval notation is negative 6 to negative 3 within parentheses.

381.

The solution is negative 2 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 2 and an open circle at 2 with shading between the closed and open circles. Its interval notation is negative 2 to 2 within a bracket and a parenthesis.

383.

The solution is negative 1 is less than x which is less than or equal to three-halves. Its graph has an open circle at negative 1 and a closed circle at three-halves with shading between the open and closed circles. Its interval notation is negative 1 to three-halves within a parenthesis and a bracket.

385.

The solution is negative 2 is less than x which is less than 2. Its graph has an open circle at negative 2 and an open circle at 2 with shading between the open circles. Its interval notation is negative 2 to 2 within parentheses.

387.

The solution is x is less than negative 2. Its graph has an open circle at negative 2 and is shaded to the left. Its interval notation is negative infinity to negative 2 within parentheses.

389.

The solution is x is greater than or equal to negative 2. Its graph has a closed circle at negative 2 and is shaded to the right. Its interval notation is negative 2 to infinity within a bracket and a parenthesis.

391.

The solution is x is less than or equal to 12. Its graph has a closed circle at 12 and is shaded to the left. Its interval notation is negative infinity to 12 within a parenthesis and a bracket.

393.

The solution is a contradiction. So, there is no solution. As a result, there is no graph or the number line or interval notation.

395.

397.

The solution is 1 is less than x which is less than or equal to 3. Its graph has an open circle at 1 and a closed circle at 3 and is shaded between the open and closed circles. Its interval notation is 1 to 3 within a parenthesis and a bracket.

399.

The solution is negative 1 is less than x which is less than 2. Its graph has an open circle at negative 1 an open circle at 2 and is shaded between. Its interval notation is negative 1 to 2 within parentheses.

401.

The solution is negative 1 is less than or equal to x which is less than or 0. Its graph has a closed circle at negative 1 and an open circle at 0 and is shaded between the closed and open circles. Its interval notation is negative 1 to 0 within a bracket and a parenthesis.

403.

The solution is x is less than or equal to negative 4 or x is greater than negative 3. The graph of the solutions on a number line has a closed circle at negative 4 and shading to the left and an open circle at negative 3 with shading to the right. The interval notation is the union of negative infinity to negative 4 within a parenthesis and a bracket and negative 3 and infinity within parentheses.

405.

The solution is x is less than 0 or x is greater than or equal to 2. The graph of the solutions on a number line has an open circle at 0 and shading to the left and a closed circle at 4 with shading to the right. The interval notation is the union of negative infinity to 0 within parentheses and 4 to infinity within a bracket and parenthesis.

407.

The solution is x is less than or equal to negative 2 or x is greater than or equal to 2. The graph of the solutions on a number line has a closed circle at negative 2 and shading to the left and a closed circle at 2 with shading to the right. The interval notation is the union of negative infinity to negative 2 within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.

409.

The solution is x is less than two-thirds or x is greater than 1. The graph of the solutions on a number line has an open circle at two-thirds and shading to the left and an open circle at 1 with shading to the right. The interval notation is the union of negative infinity to two-thirds within parentheses and 1 and infinity within parentheses.

411.

The solution is x is less than 3 or x is greater than 12. The graph of the solutions on a number line has an open circle at 3 and shading to the left and an open circle at 12 with shading to the right. The interval notation is the union of negative infinity to 3 within parentheses and 12 and infinity within parentheses.

413.

415.

417.

The solution is x is less than 1. Its graph has an open circle at negative 1 is shaded to the right. Its interval notation is 1 to infinity within parentheses.

419.

421.

423.

The solution is negative 1 is less than x which is less than or equal to two-thirds. Its graph has an open circle at negative 1 and a closed circle at two-thirds and is shaded between the open and closed circles. Its interval notation is negative 1 to two-thirds within a parenthesis and a bracket.

425.

The solution is negative 1 is less than or equal to x which is less than 2. Its graph has a closed circle at negative 1 and a closed circle at 2 and is shaded between the closed circles. Its interval notation is negative 1 to 4 within brackets.

427.

$5 \leq n \leq 24$

429.

$6 \leq w \leq 12$

431.

ⓐ answers vary ⓑ answers vary

433.

Answers will vary.

Section 2.7 Exercises

435.

437.

439.

$x = 1, x = - \frac{1}{2}$

441.

$x = −1, x = - \frac{5}{2}$

443.

$x = 7, x = 1$

445.

$x = 1, x = −5$

447.

$x = 7, x = 1$

449.

no solution

451.

no solution

453.

$x = −1, x = - \frac{2}{3}$

455.

$x = −3, x = 3$

457.

The solution is negative 1 is less than x which is less than 1. The number line shows an open circle at negative 1, an open circle at 1, and shading between the circles. The interval notation is negative 1 to 1 within parentheses.

459.

The solution is negative 3 is less than or equal to x which is less than or equal to 3. The number line shows a closed circle at negative 3, a closed circle at 3, and shading between the circles. The interval notation is negative 3 to 3 within brackets.

461.

The solution is 1 is less than or equal to x which is less than or equal to 4. The number line shows a closed circle at 1, a closed circle at 4, and shading between the circles. The interval notation is 1 to 4 within brackets.

463.

465.

The solution is negative one-third is less than x which is less than 2. The number line shows an open circle at negative one-half, an open circle at 2, and shading between the circles. The interval notation is negative one-third to 2 within parentheses.

467.

469.

The solution is x is less than negative 6 or x is greater than 6. The number line shows an open circle at negative 6 with shading to its left and an open circle at 6 with shading to its right. The interval notation is the union of negative infinity to negative 6 within parentheses and 6 to infinity within parentheses

471.

The solution is x is less then or equal to negative 5 or x is greater than or equal to 5. The number line shows an open circle at negative 5 with shading to its left and an open circle at 5 with shading to its right. The interval notation is the union of negative infinity to negative 5 within parentheses and 5 to infinity within parentheses.

473.

475.

The solution is x is less than negative 2 or x is greater than 3. The number line shows an open circle at negative 2 with shading to its left and an open circle at 3 with shading to its right. The interval notation is the union of negative infinity to negative 2 within parentheses and 3 to infinity within parentheses.

477.

The solution is x is less than or equal to 6 or x is greater than or equal to 8. The number line shows a closed circle at 6 with shading to its left and a closed circle at 8 with shading to its right. The interval notation is the union of negative infinity to 6 within parenthesis and a bracket and 8 to infinity within a bracket and a parenthesis.

479.

481.

$x = 2, x = \frac{1}{4}$

483.

$x = 3, x = 2$

485.

$x = 3, x = - \frac{11}{3}$

487.

$x = \frac{3}{2}, x = - \frac{1}{2}$

489.

491.

The minimum to maximum expected production is 207,500 to 2,225,000 bottles

493.

The acceptable weight is 22.5 to 25.5 ounces.

495.

Answers will vary.

Review Exercises

497.

no

499.

$s = −22$

501.

$m = −14$

503.

$q = 18$

505.

$x = −1$

507.

$k = \frac{3}{4}$

509.

contradiction; no solution

511.

$n = 2$

513.

$k = 23$

515.

$x = 5$

517.

There are 116 people.

519.

38

521.

$−3, −10$

523.

76, 78, 80

525.

$922

527.

160

529.

$$ 3.89$

531.

35%

533.

32%

535.

ⓐ $105 ⓑ $52.5 %$

537.

$428.22

539.

$1,900

541.

$d_{2} = \frac{2 A}{d_{1}}$

543.

$y = \frac{4 x}{3} - 4$

545.

$22.5 °, 67.5 °, 90 °$

547.

26

549.

6 feet

551.

$24.5$ cm, $12.5$ cm

553.

9 ft, 14 ft, 12 ft

555.

nine pennies, six dimes, 12 quarters

557.

57 students, 68 general admission

559.

$2.2$ lbs of raisins, $7.8$ lbs of nuts

561.

$9.7 %$

563.

Louellen 65 mph, Tracy 66 mph

565.

upstream 3 mph, downstream 5 mph

567.

heavy traffic 32 mph, light traffic 68 mph

569.

The solution is x is greater than or equal to negative 2.5. The number line shows a left bracket at negative 2.5 with shading to its right. The interval notation is negative 2.5 to infinity within a bracket and a parenthesis.

571.

573.

The solution is negative 5 is less than or equal to x which is less than negative 3. The number line shows a closed circle at negative 5, an open circle at negative 3, and shading between the circles. The interval notation is negative 5 to negative 3 within a bracket and a parenthesis.

575.

The solution is n is less than or equal to 35. The number line shows a a right bracket at 35 with shading to its left. The interval notation is negative infinity to 35 within a parenthesis and a bracket.

577.

The solution is x is greater than 6. The number line shows a left parenthesis at 6 with shading to its right. The interval notation is 6 to infinity within parentheses.

579.

The solution is p is less than ten-thirds. The number line shows a right parenthesis at ten-thirds with shading to its left. The interval notation is negative infinity to ten-thirds within parentheses.

581.

583.

The inequality is z plus 5 is less than or equal to 19. Its solution is z is less than or equal to 14. The number line shows a right bracket at 14 with shading to its left. The interval notation is negative infinity to 14 within a parenthesis and a bracket.

585.

The inequality is 9 n is greater than 42. Its solution is n is greater than fourteen-thirds. The number line shows a left parentheses at fourteen-thirds with shading to its right. The interval notation is fourteen-thirds to infinity within parentheses.

587.

$33 per day

589.

at least $300,000

591.

at least 112 jobs

593.

The solution is negative 3 is less than x which is less than or equal to 5. The number line shows an open circle at negative 3 and a closed circle at 5. The interval notation is negative 3 to 5 within a parenthesis and a bracket.

595.

The solution is negative x is less than negative five-fourths. The number line shows an open circle at negative five-fourths with shading to its left. The interval notation is negative infinity to negative five-fourths within parentheses.

597.

The solution is a contradiction. So, there is no solution. As a result, there is no graph on the number line or interval notation

599.

The solution is x is less than negative two-thirds or x is greater than or equal to 3. The number line shows a closed circle at negative two-thirds with shading to its left and a closed circle at 3 with shading to its right. The interval notation is the union of negative infinity to negative two-thirds within a parenthesis and a bracket and 3 to infinity within a bracket and a parenthesis.

601.

The solution is x is less than 2 or x is greater than 8. The number line shows an open circle at 2 with shading to its left and an open circle at 8 with shading to its right. The interval notation is the union of negative infinity to 8 within parentheses and 8 to infinity within parentheses.

603.

605.

$6 \leq w \leq 12$

607.

no solution

609.

$x = 2, x = \frac{2}{3}$

611.

$x = 9, x = −3$

613.

$x = 2, x = \frac{1}{4}$

615.

The solution is 1 is less than or equal to x which is less than or equal to 4. The number line shows a closed circle at 1, a closed circle at 4, and shading in between the circles. The interval notation is 1 to 4 within brackets.

617.

619.

The solution is x is less than negative 2 or x is greater than 6. The number line shows a closed circle at negative 2 with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative 2 within a parenthesis and a bracket and 2 to infinity within a bracket and a parenthesis.

621.

623.

The minimum to maximum expected usage is 210,000 to 220,000 bottles

Practice Test

625.

$x = −5$

627.

$a = 41$

629.

contradiction; no solution

631.

$x = 6$

633.

$x = −2, x = - \frac{1}{3}$

635.

637.

639.

The solution is c is greater than one-third. The number line shows a left parenthesis at one-third with shading to its right. The interval notation is one-third to infinity within parentheses.

641.

The solution is x is less than two-thirds or x is greater than 1. The number line shows an open circle at two-thirds with shading to its left and an open circle at 1 with shading to its right. The interval notation is the union of negative infinity to two-thirds within parentheses and 1 to infinity within parentheses.

643.

The solution is x is less than or equal to negative one-half or x is greater than or equal to 2. The number line shows a closed circle at negative one-half with shading to its left and a closed circle at 2 with shading to its right. The interval notation is the union of negative infinity to negative one-half within a parenthesis and bracket and 2 to infinity within a bracket and a parenthesis.

645.

$10.8$

647.

$−57, −55$

649.

12 dimes, seven quarters

651.

$30 °, 60 °, 90 °$

653.

$2.5$ hours

655.

At most $55.56 per costume.