Chapter 9
Inequalities and Problem Solving
9.1 Check Points
1.
4323
x
2.
31715
416
416
44
4
xx
x
x
x
,4
3.
425
236
13
xx
x
13,
4. a.
() () ()
( ) 200 (160,000 75 )
Px Rx Cx
Px x x
5. a. () 300,000 30Cx x
d.
() 0
50 300,000 0
50 300, 000
50 300,000
50 50
6000
Px
x
x
x
x
For the business to make money, more than
6000 pairs must be produced and sold.
9.1 Exercise Set
1.
51126
515
x
x
Chapter 9 Inequalities and Problem Solving
3.
3813
321
7
x
x
x
The solution set is
7, .
5.
936
4
x
x
7.
811313
511 13
52
2
xx
x
x
x
8.
18 45 12 8
6458
653
xx
x
x
9.
41236
44236
4636
66
0
xx
xx
xx
x
x
38
5
x
x
The solution set is
5,
.
The solution set is
,1 .
12.
42320
48320
xx
xx
13.
1342
xx
Section 9.1 Reviewing Linear Inequalities and Using Inequalities in Business Applications
14.
53 3 1
xx
15.
11
42 2
1
44441
422
xx
xx
16.
31
1
10 5 10
31
10 10 1 10 10
10 5 10
xx
xx
17.
14
2
21 2 24
2
x
x
18.
43
755
x
The solution set is
8,
.
19.
425
6918
xx
The solution set is
13,
.
20.
43 21
2
612
xx
The solution set is 19 ,
6
.
Chapter 9 Inequalities and Problem Solving
21.
7 4 13 12 13 3
72813123913
7151351
xx
xx
xx
22.
37 2 3 2 1
37 2 3 2 2
xx x
xx x
23. 22
6 (3 12) (10 50)
35
62 84 20
xx
xx
24.
23
7 21 4 10 11 11
711
26 4103 3
xx
xx
Section 9.1 Reviewing Linear Inequalities and Using Inequalities in Business Applications
25.
33 5 8 7 53 6 23 5 24 3
3 3 15 8 7 5 3 18 6 10 8 6
xx x x x
xx xx x
26.
532 3 25 65 2 24 3 3 19xxx xx
56 9 10 2 65 10 8 6 3 19
xxxxx
27. () ()
3258
22 8
fx gx
xx
x
28. 2954
394
313
xx
x
x
Chapter 9 Inequalities and Problem Solving
29.
() ()fx gx
12
812 10 15
45
23 4 6
xx
xx
30.
33
16 8 7 10 15 9
85
63 7 6 99
316
xx
xx
xx
9
31. 1( 3)2 4
1324
xx
xx
32. 2113(2)0
xx
33. a.
32 25,500 15
32 25,500 15
Px Rx Cx
xx
xx
More than 1500 units must be produced and sold
to have a profit.
34. a.
32 15,000 12
Px Rx Cx
xx
20 15, 000
20 20
750
x
x
35. a.
245 105 70,000
Px Rx Cx
xx
More than 500 units must be produced and sold
to have a profit.
Section 9.1 Reviewing Linear Inequalities and Using Inequalities in Business Applications
36. a.
Px Rx Cx
b.
0
0.5 1500 0
Px
x
37.
2 364 34 4xxxx
2664 34 4
2664344
xxxx
xx
The solution set is { | 2} or (- ,2).xx
38.
346456432xxxxx
12 18 4 5 6 4 3 2
12 18 4 5 6 2
xxxxx
xxxx
39.
,0
ax b c a
aa
cb
xa
xa
41. (,3]
42. [3, )
intimacy
passion
48. commitment
intimacy or
intimacy
commitment
49.
passion < commitment or
commitment > passion
50. commitment > passion or
passion < commitment
Chapter 9 Inequalities and Problem Solving
b.
0.7 34
Dx
54. a.
0.7 34
0.7 20 34
48
Dx
D
D
The formula gives 48% as the national diversity
index in 2000. This overestimated the actual
diversit index from the graph by 1%.
55. a. The slope of the model is –8. This represents
that per-capita consumption of carbonated soda
is decreasing by eight 12-ounce servings per
year on average after 1998.
56. a. The slope of the model is –8. This represents
that per-capita consumption of carbonated soda
is decreasing by eight 12-ounce servings per
year on average after 1998.
57. a.
cost fixed costs variable cost
c.
80 18,000 20
80 18,000 20
60 18,000
Px Rx Cx
xx
xx
x
d.
0
Px
More than 300 canoes need to be produced and
sold in order to make a profit.
58. a.
cost fixed costs variable cost
100,000 100Cx x
200 100,000
x
d.
0
200 100,000 0
Px
x
Section 9.1 Reviewing Linear Inequalities and Using Inequalities in Business Applications
59. a.
cost overhead per show cost
30,000 2500Cx x
d.
0
625 30,000 0
Px
x
60. a.
cost overhead per card cost
30,000 0.02Cx x
b.
revenue price quantity
0.50Rx x
d.
0
0.48 30,000 0
0.48 30,000
0.48 30,000
0.48 0.48
62,500
Px
x
x
x
x
61.
The cost is
10,000 0.40 .Cx
The revenue is
2.
R
x
CR
62. The cost is
3000 3 .Cx
The revenue is
5.5 .
R
x
CR
sold each week.
63. The cost with Plan A is
15 0.08 .
A
Cx
The cost with Plan B is
30.12.
B
Cx
15 0.08 3 0.12
15 0.04 3
AB
CC
xx
x
64.
The tax bill assessed under the first tax bill is
11800 0.03 .Tx
The tax bill assessed under the second tax bill is
2200 0.08 .Tx
12
1800 0.03 200 0.08
1800 0.05 200
TT
xx
x
Chapter 9 Inequalities and Problem Solving
70.
3622
xx
71.
24616
xx
72. Exercise 70
73. a. Plan A:
40.10
x
Plan B:
20.15
x
b. Window: [0,50,1] by [0,10,1]
checks per month.
d.
40.10 20.15
4 2 0.05
0.05 2 4
AB
xx
x
x
Sample explanation: The statement “three times a
number is less than two times a number” is true for
negative numbers.
77. makes sense
78. false; Changes to make the statement true will vary.
A sample change is:
36
x
is equivalent to
2,
x
not
2
x
.
Section 9.1 Reviewing Linear Inequalities and Using Inequalities in Business Applications
82. Find a when
8.
x
412
(8) 4 12
816
ax
a
a
83.
Since
,
xy
then 0.yx When multiplying
both sides of the inequality by
yx
, remember
to flip the inequality.
21
84.
2
4 4 2 4 51685 29f
85. Add the first and third equations to eliminate y.
2 3
2 4
1
xyz
xyz
xz
Multiply the third equation by 2 and add to the
second equation.
1
21
1
xz
x
x
Back-substitute 2 for z and –1 and x in one of the
The solution set is
1, 1, 2 .
86.
2
25 81 5 9 5 9xxx
87. a. {3, 4}
b.
2414
210
5
x
x
x
The solution set is
5or ,5.xx
c. Answers will vary. Any number less than 5.
d. Answers will vary. Any number in
1, 3 .
Chapter 9 Inequalities and Problem Solving
9.2 Check Points
3,4,5,6,7 3,7,8,9 3,7
2. Solve and graph each inequality, and graph the
intersection.
25 and 2 4 2
xx
The solution set is
,1 .
3. Solve and graph each inequality, and graph the
intersection.
457 and523
xx
Since the two sets do not intersect, the solution set
is .
4. 12 311
13 2 33113
x
x
5.
3, 4, 5, 6, 7 3, 7,8, 9 3, 4, 5, 6, 7,8, 9
6. Solve and graph each inequality, and graph the
intersection.
33 2 6
126
22
3
xx
xx
x
7. Solve and graph each inequality, and graph the
intersection.
253 or 233
22 20
10
xx
xx
xx
9.2 Concept and Vocabulary Check
1. intersection;
AB
9.2 Exercise Set
Section 9.2 Compound Inequalities
3.
1, 3, 5, 7 2, 4, 6, 8,10 or
7.
3 and 6
xx
The solution set is
6, .
9.
5 and 1
xx
The solution set is
,1 .
11.
2 and 1
xx
12. 3 and 1
xx
13.
2 and 1
xx
Since the two sets do not intersect, the solution set
is .
15.
520and318
46
xx
xx
The solution set is
3, 5.
Chapter 9 Inequalities and Problem Solving
17. 42and3 1 8
639
3
xx
xx
x
18. 32 4and215
36 26
23
xx
xx
xx
19. 2515and7210
315 510
52
xx xx
xx
xx
20. 65 13 and 4 3 9
62 1 3 3 9
2
xx xx
xx
21.
7
41 6 and 2
5
44 6 7
552
410 5
5710
x
x
xx
x
Since the two sets do not intersect, the solution set
is .
and 6
x
Since the two sets do not intersect, the solution set
is .
02
0
xx
x
The solution set is
0, 2 .
24. 2143and 135
21 3 215
xx xx
xx
Section 9.2 Compound Inequalities
25. 638
63 3383
35
x
x
x
27. 321
32 2212
13
x
x
x
The solution set is
1, 3.
29. 11 2 1 5
11 1 2 1 1 5 1
10 2 4
52
x
x
x
x
The solution set is
5, 2 .
31.
2
351
3
2
x
x
The solution set is
3, 6
.
32. 643
2
21
2
x
x
34.
1, 3, 7, 8 2, 3, 8 1, 2, 3, 7, 8
35.
1, 3, 5, 7 2, 4, 6, 8,10
1, 2,3,4,5,6,7,8,10
36.
0,1, 3, 5 2, 4, 6 0,1, 2, 3, 4, 5, 6
Chapter 9 Inequalities and Problem Solving
40. 2 or 4xx
41. 5 or 1xx
42. 6 or 2xx
43. 2 or 1xx
44. 3 or 1xx
45. 2 or 1xx
46. 3 or 1xx
47. 312or2 6
43
xx
xx
The solution set is
,3 4,
.
48. 33or210
xx
The solution set is
,1 5,
.
26 55
31
xx
xx
The solution set is
,3 1,
.
Section 9.2 Compound Inequalities
51. 43 1or 23 11
44 28
14
xx
xx
xx
52. 2115or34 1
214 33
71
xx
xx
xx
53. 257or3102
22 5100
1510
2
xxx
xx
xx
x
54. 16 3 8 or 13 4 3
324 1353
8510
2
xxx
xx
xx
x
55. 235and3111
22 312
14
xx
xx
xx
The solution set is
4, .
56. 455and342
40 36
02
xx
xx
xx
The solution set is
0, 2 .
57. 31 1or4 2
30 42
06
xx
xx
xx
The solution set is
,0 6,
.
58. 253or3 0
xx
The solution set is
3, .
Chapter 9 Inequalities and Problem Solving
59. 0, 0, 0abc
60. 0, 0, 0abc
22
ax b
c
62.
1, 3
63. Solving in separate pieces:
22 1
xx
and 21 2
xx
64. Solving in separate pieces:
310
02 10
10 2
5
xx
x
x
x
and 3102
10 0
10
xx
x
x
The solution set is
5,10 .
65. The solution set is
1, 2.
68. 53 416
93 12
34
x
x
x
69. a.
1
26Ix
More than 33% of U.S. households will have an
interfaith marriage in years after 2016 (i.e.
1988 28).
b.
1
6Nx
c. More than 33% of U.S. households will have an
interfaith marriage and more than 14% of U.S.
households will have a person of faith married to
someone with no religion in years after 2020.
70. a.
1
4
26Ix
1
4
1
4
26 34
8
32
x
x
x
More than 34% of U.S. households will have an
interfaith marriage in years after 2020 (i.e.
1988 32).
c. More than 34% of U.S. households will have an
interfaith marriage and more than 15% of U.S.
households will have a person of faith married to
someone with no religion in years after 2020.
Section 9.2 Compound Inequalities
71.
9
41 32 50
5
9
41 32 32 32 50 32
C
C
72.
5
15 32 35
9
995 9
15 32 35
559 5
F
F
73. Let x = the score on the fifth exam.
70 75 87 92
80 90
5
324
80 90
5
x
x
A grade between 76 and 125 is needed on the fifth
exam.
Because the inequality states the score must be less
than 126, we say 125 is the highest possible score.
74. Let x = the score on the final exam.
82 75 80 90 2
80 90
6
327 2
x
x
75. Let x = the number of times the bridge is crossed
per three month period.
The cost with the 3-month pass is
3
7.50 0.50 .Cx
15
x
We also must consider the cost without purchasing
a pass. We need this cost to be less than the cost
with a 3-month pass.
37.500.50
xx
76. Let x = the number of hours the mechanic works on
the car.
226 175 34 294
51 34 119
x
x
Chapter 9 Inequalities and Problem Solving
83. 139x
Using the intersection feature, find the range of the
x-values of the points lying between the two
constant functions.
84. 4
13
2
x
We need to find the range of the x-values of the
points lying between the two constant functions.
Using the intersection feature, we can determine the
x-values of the endpoints of the range.
85. 14 73x
Using the intersection feature, find the range of the
x-values of the points lying between the two
constant functions.
86. 24 7x
We need to find the range of the x-values of the
points lying between the two constant functions.
Using the intersection feature, we can determine the
x-values of the endpoints of the range.
87.a.
c.