Section 9.2 Compound Inequalities
88. makes sense
89. makes sense
93. false; Changes to make the statement true will vary.
A sample change is:
,3 , 2 ,3
94. false; Changes to make the statement true will vary.
97. The domain of
,4 .f
98. The domain of g =
1,.
100. The domain of
1, 4 .
f
g
101. Let n = number of nickels.
Let d = number of dimes.
Let q = number of quarters.
320 5 10 25 545
nd q
102.
2
25 34
gfx gx fx
xxx
159 15
103.
42 8
248
24
xy
yx
yx
104.
42 454285
42 13
42 13
17 2
xx
x
x
x
31 5 2 6
44 3
1
xx x
xx
x
107. a.
235
2( 5) 3 5
x
Chapter 9 Inequalities and Problem Solving
b.
235
x
9.3 Check Points
1.
215x
Rewrite without absolute value bars.
2. First, isolate
13.x
21 3 28 0
21 3 28
13 14
x
x
x
Rewrite without absolute value bars.
3.
27 3xx
27 3or 27 (3)
xx x x
4.
25x
5. First, isolate the absolute value expression on one
side of the inequality.
35 2 20 19
x
Rewrite without absolute value bars.
uc means .cuc
13 5 2 13
13 2 5 2 2 13 2
11 5 15
11 5 15
555
x
x
x
x
uc means or .uc uc
253x
253or 253
xx
Section 9.3 Equations and Inequalities Involving Absolute Value
7. 41 3.2x
Rewrite without absolute value bars.
uc means .cuc
9.3 Concept and Vocabulary Check
1. c; c
2. v; v
3. c; c
10. B
11. D
12. F
9.3 Exercise Set
1. 8x
8or 8xx
3. 27
27or 2 7
95
x
xx
xx
5.
217
217or21 7
28 2 6
43
x
xx
xx
xx
The solutions set is
3, 4 .
6. 2311
x
4232 4232
426 426
48 4 4
21
xx
xx
xx
xx
The solutions set is
1, 2 .
8.
31
1
5
x
31 31
1or 1
xx
3
Chapter 9 Inequalities and Problem Solving
9. 8x
The solution set is . There are no values of x for
which the absolute value of x is a negative number.
12. 20
20
2
x
x
x
The solution set is
2.
14. 3512
54
y
y
54 or 5 4
19
yy
yy
The solution set is
9, 1 .
16. 23 2 14
327
x
x
630 6 26
526
6
13
3
yy
yy
y
16 14
33
yy
The solution set is 14 16
,.
33
19. 75 2 16
x
Section 9.3 Equations and Inequalities Involving Absolute Value
21. 153
12
x
x
The solution set is . By definition, absolute
values are always zero or positive.
24. 3281
32 7
y
y
The solution set is . By definition, absolute
values are always zero or positive.
The solutions set is 1.
2
27. 5832xx
5832or
58 32
282 88 2
210 86
xx xx
xx
xx
568
84
63
xx
x
The solution set is 4,5 .
3
3
30. 639xx
639or
639
xx xx
Chapter 9 Inequalities and Problem Solving
31. 2525xx
2525 or25 25
xx x x
32. 3535xx
35 35
3535 or
xx
xx
33. 35xx
35 or 35
xx
xx
34. 36xx
36 or 3 6
xxx x
2
35. 26102yy
36. 4345yy
43 45
4345or
yy
yy
37. 223
33
xx
15
x
or
2
3323 33
33
xx
38. 1
2
22
xx
11
2or 2
xx
xx
Section 9.3 Equations and Inequalities Involving Absolute Value
39. 3
33
x
x
40. 5
55
x
x
The solution set is
5, 5.
43. 21
121
12 2212
31
x
x
x
x
The solution set is
3, 1 .
45. 268
82 68
x
x
The solution set is
1, 7 .
46. 3517
17 3 5 17
x
x
48. 5x
5or 5xx
The solution set is
,5 5, .
49. 31x
31or 31
xx
Chapter 9 Inequalities and Problem Solving
52. 34x
34or 34
17
xx
xx
The solution set is
,1 7,
.
54. 5213x
52 13or5213
xx
55.
2 148
2248
228
x
x
x
19 7
3
x
The solution set is
19 ,7
3
.
26
y
The solution set is
6, 0 .
79
x
The solution set is
7, 9 .
Section 9.3 Equations and Inequalities Involving Absolute Value
60.
33
1
x
61.
2
35
3
x
22
or
3535
33
xx
62.
3
39
4
x
33
or
3939
44
xx
63.
21x
The solution set is
.
Since all absolute values are
zero or positive, there are no values of x that will
make the absolute value of the expression less than
–1.
64.
32x
65.
610x
66.
412x
Since all absolute values are zero or positive, we
know that when simplified, the left hand side will
be a positive number. We also know that any
positive number is greater than any negative
727
72 2272
95
x
x
x
13
x
The solution set is
1, 3
.
69.
22 3 10 12
22 3 2
x
x
Chapter 9 Inequalities and Problem Solving
70.
32 1 2 8
32 1 6
212
x
x
x
71.
41 16
41 16
44
14
x
x
x
72.
25 6
25 6
22
53
x
x
x
73.
32 1
213
x
x
74.
94 7
479
x
x
479or47 9
xx
75.
54 11x
5 4 11 or 5 4 11
46 4 16
34
xx
xx
y
x
311 3 15
11 5
3
xx
x
x
The solution set is
11 ,5 .
2214
7
xx
x
The solution set is
7, 2 .
Section 9.3 Equations and Inequalities Involving Absolute Value
79.
|13 1|5x
51335
x
80.
34 1 3
34 4 3
x
x
81.
|2 3| 1 6
|2 3| 5
x
x
82.
24618
2424
x
x
83. Let x be the number.
|4 3 | 5x
84. Let x be the number.
|5 4 | 13x
85. ||ax b c
When solving, we do not reverse the inequality
symbol from “<” to “>” when dividing by a
86. ||ax b c
ax b c
or
ax b c
87.
41x
The graphs of
4fx x
and 1y intersect
The graphs of
4fx x
is below the graph of
5y when x is between
1
and 9. Thus, the
Chapter 9 Inequalities and Problem Solving
91.
21 3
3213
321 2121321
x
x
x
92.
15 3
3153
x
x
93.
57 7
7577
T
T
94.
50 22
22 50 22
28 72
T
T
T
The monthly average temperature for Albany, New
York ranges from
28 F
to
72 F
, inclusive.
95.
8.6 0.01
x
96.
9.4 0.01
0.01 9.4 0.01
9.39 9.41
x
x
x
97.
50 1.645
5
h
50 1.645
h
50 1.645
h
resulted in 41 or less heads, or 59 or more heads.
98. – 104. Answers will vary.
105.
15x
The solutions are –6 and 4 and the solution set is
The solutions are –8 and 0 and the solution set is
Section 9.3 Equations and Inequalities Involving Absolute Value
108.
235x
109.
215
33
x
110.
41x
111.
217x
112.
0.1 0.4 0.4 0.6x
113.
41x
114. Answers will vary.
115. does not make sense; Explanations will vary.
Sample explanation:
25x
means
25or 25.xx .
see that the absolute value expression must be less
than 5. This is possible so the inequality does have a
solution.
118. makes sense
119. false; Changes to make the statement true will vary.
A sample change is: Some absolute value equations,
Chapter 9 Inequalities and Problem Solving
123. a.
43x
b.
43x
124. The proportion of refunds will be
0.3% 0.2%
0.003 0.002
p
p
125. |2 5| 3 4xx
2534
54
1
1
xx
x
x
x
or
25 34
25 34
55 4
59
9
xx
xx
x
x
x
128.
2fx
is the horizontal line positioned at
2.y
Mid-Chapter Check Point
2.
52 19
62 10
35
x
x
x
The solution set is
3, 5 .
4.
10 3 2 1 8 1
10 6 3 8 1
61381
13 14 1
14 14
1
1
xx
xx
xx
x
x
x
x
The solution set is
,1.
7.
|5||58|xx
55 8
458
xx
x
or
558
558
xx
xx
8.
52 9
52 9
42
2
x
x
x
x
and
53 17
520
4
x
x
x
10.
5
3
232
5
636
232
318215
18 15
3
xx
xx
xx
x
x
The solution set is
,3.
1
5
x
1
x
The solution set is
1
,1 ,
5
.
Chapter 9 Inequalities and Problem Solving
13.
724
2
x
14.
31
43
394
35
5
3
x
x
x
x
The solution set is
5
,.
3
16.
32 5 6
25 9
259
x
x
x
259
214
7
x
x
x
or
259
24
2
x
x
x
The solution set is
2,7
.
18. a.
cost fixed costs variable cost
60, 000 0.18Cx x
0.12 60,000
x
d. Let x = number of compact discs.
0.30 60,000 0.18 30,000
0.30 60,000 0.18 30,000
0.12 60, 000 30,000
0.12 90,000
750,000
xx
xx
x
x
x
The company should produce and sell at least
750,000 compact discs.
20. Let x = grade on the fifth exam.
95 79 91 86
80 90
5
351
80 90
5
400 351 450
49 99
x
x
x
x
49,99
Section 9.4 Llnear Inequalities In Two Variables
9.4 Check Points
1.
42 8xy
First, graph the equation 42 8xy
with a solid
line.
Set 0y to find the x -intercept.
() 0
Px
6, .
4
y
Next, use the origin as a test point.
42 8
4(0) 2(0) 8
08,false
xy
Since the statement is false, shade the half-plane
2.
3
4
yx
First, graph the equation
3
4
yx
with a dashed
line.
3(, )
4
xy x xy
3
4
3
4(0)
4
40,true
yx
Since the statement is true, shade the half-plane that
Since the inequality is of the form
,ya
shade
the half-plane above the line.
Chapter 9 Inequalities and Problem Solving
4. Point (60,20).B Check this point in each of the three inequalities for grasslands.
35 5 7 70 3 35 140
TTPTP
5.
36
23 6
xy
xy
Graph the line 36xy
with a dashed line. Graph the line 23 6xy
with a solid line.
For 36xy
use a test point such as (0, 0).
The solution set of the system is the intersection (the overlap) of the two half-planes.
6.
2
21
3
xy
x
y
Graph the line 2xy with a dashed line.
Graph the line
2x
with a solid line.
Section 9.4 Llnear Inequalities In Two Variables
9.4 Concept and Vocabulary Check
1. solution; x; y;
51
2. graph
3. half-plane
9.4 Exercise Set
1. 3xy
First, graph the equation 3xy. Rewrite the
equation in slope-intercept form by solving for y.
3
xy
2.
3. 5xy
First, graph the equation 5xy. Rewrite the
equation in slope-intercept form by solving for y.
5
5
5
xy
yx
yx
y -intercept = –5
slope = 1
4.
Chapter 9 Inequalities and Problem Solving
5. 24xy
First, graph the equation 24xy
. Rewrite the
equation in slope-intercept form by solving for y.
24
24
xy
yx
7.
36xy
First, graph the equation
36xy
. Rewrite
the equation in slope-intercept form by solving
for y.
36
36
xy
yx
8.
32 6
236
33
2
xy
yx
yx
y-intercept = 3 slope =
3
2
Next, use the origin as a test point.