⫺40
⫺9
⫺70
Chapter 9
More on Inequalities
Exercise Set 9.1
RC6. (d)
2. 3x+5≤−10
0: 3·0+5≤−10, or 5 ≤−10 is false.
4. 5y−7<8−y
2: 5·2−7<8−2, or 3 <6 is true.
2 is a solution.
6. [−5,∞)
8. (−10,10]
y<6
20. a+6≤−14
a≤−20
22. x−8≤17
24. y−9>−18
26. y−18 ≤−4
28. 8t<−56
30. 0.6x<30
34. −5y≤3.5
240 Chapter 9: More on Inequalities
36. −1
9
38. 5y+13>28
40. −9x+3x≥−24
46. 2x−3<13
4x+10−1.25x
48. 2m+5≥16(m−4)
50. 2(0.5−3y)+y>(4y−0.2)8
1−6y+y>32y−1.6
52. [8x−3(3x+ 2)] −5≥3(x+4)−2x
23
54. 5(t+3)+9<3(t−2)+6
56. 13 −(2c+2)≥2(c+2)+3c
5x>30
5[−11 −11t]−20 ≤−6[8+6t]
62. 2
3(4x−3) >30
Exercise Set 9.1 241
64. 7
8(5 −4x)−17 ≥38
66. 2
37
8−4x−5
8<3
8
68. 0.9(2x+8)<20 −(x+5)
x< 78
28,or 39
14
70. 0.8−4(b−1) >0.2 + 3(4 −b)
−10b>74
72. 703W
772<25
74. Let x= the score on the fourth test. It is possible to score
76. Let m= the number of miles for which PDQ is less expen-
sive. Solve:
78. 12.50n>300+9n
3.5n>300
250+0.1b−25 <50+0.2b−10
185 <0.1b
82. Let x= the amount invested at 3%.
−0.01x≥−150
84. a) 5
In terms of an inequality we write
F<1761.44
86. Let d= the dewpoint spread. Then d
20
d
88. (r−4s)(6r+s)=6r2−23rs −4s2
96. f(x)=3x−5
x=2
3
100. False; −3<−2, but (−3)2>(−2)2.
102. No. Let x= 2. Then x<3 is true, but 0 ·x<0·3, or
Exercise Set 9.2
8. {m, n, o, p}∪{m, o, p}={m, n, o, p}
16. Interval notation for −3≤y≤4 is [−3,4].
−8<4x and 4x≤16
20. 4x−7<1and 7−3x>−8
22. 5−7x>19 and 2−3x<−4
∅
24. −6<x+6≤8
28. −6≤x+1<9
30. 5≤8x+5≤21
32. −6≤2x−3<6
⫺660⫺5⫺4⫺3⫺2⫺1 54321
Exercise Set 9.2 243
34. 4>−3m−7≥2
36. −2
3≤4−1
4x< 2
3
38. −3<2x−5
4<8
−7<2x<37
40. x<−4or x > 0 can be written in interval notation as
(−∞,−4) ∪(0,∞).
42. x≤−1or x > 3 can be written in interval notation as
44. x−2<−1or x −2>3
46. x−5≤−4or 2x−7≥3
48. 4x−4<−8or 4x−4<12
50. 6>2x−1or −4≤2x−1
52. 3x+2<2or 4−2x<14
54. −3m−7<−5or −3m−7>5
3,∞
56. 1
4−3x≤−3.7or 1
4−5x≥4.8
x≤−91
100 or x ≥79
60,or
−∞,−91
−3x<−27 or −3x>13
244 Chapter 9: More on Inequalities
to $100 + $2c. A one-year pass costs $400 regardless of the
64. Solve: 18.5<703W
772<24.9
156.0<W<210.0
The solution set is {W|156.0lb<W <210.0lb}.
7x+42y=−231 Multiplying (2) by 7
31x=−93
x=−3
Substitute −3 for xin (2).
70. m=7−(−4)
2−3=11
−1=−11
72. m=−2−4
74. (5y+ 6)(5y+1)=25y2+35y+6
15
y
y<34
15,or−∞,34
15
82. 2x+3≤x−6or 3x−2≤4x+5
Chapter 9 Mid-Chapter Review
1. x−5>2
2. If ais at most c, we have a≤c. We see that acan be less
than c, so the given statement is false.
−5x≤x+12
8. This is the graph of {x|−3≤x≤3},or[−3,3]. The
intersection is {0,10}.
14. {e, f, g, h}∪{b, d, e}
16. {3,6,9,12,15}∩{−12,−6,7,8}
There are no numbers common to the two sets, so the
18. −5
20
x<−9or x > 1
The solution set is {x|x<−9or x > 1},or
22. 4(3t−4) >2(6 −t)
24. 0.1y+3<5.6y−2
10(0.1y+3)<10(5.6y−2) Multiplying to
25. −6<2x−1
3<8
−18 <2x−1<24 Multiplying by 3
26. 20 −(2x−9) ≤3(x−2) + x
27. −1
246 Chapter 9: More on Inequalities
28. 2x−7>−18 or 3x−7≤40
or equal to 47
3, the solution set is {x|xis a real number},
Translate. The sum of the scores must be at least 450
points, so we have
85+96+88+95+s≥450.
Solve. We solve the inequality.
30. Familiarize. Let x= the amount invested at 4.5%. Then
12,500 −x= the amount invested at 5%. In one year
31. When the signs of the quantities on either side of the in-
7−9x+6x<−9(x+2)+10x
33. By definition, the notation 3 <x<5 indicates that 3 <x
Exercise Set 9.3
RC2. |x|≥3
2. |26x|=|26|·|x|=26|x|
24.
2
=
4
=
9
=3
Exercise Set 9.3 247
30. |y|=7.4
32. |3x−2|=6
34. |5x+2|=3
36. |9y−2|=17
38. |x|−2=6.3
|x|=8.3
x=−8.3or x =8.3
42. |2y|=18
2y=−18 or 2y=18
{−4,4}
46. 5|x|+10=26
48.
4−5x
=7
4−5x=−42 or 4−5x=42
|t−7|=9
|2x−7|=7
The absolute value of a number is always nonnegative. The
solution set is ∅.
56.
2
3−4x
=4
5
30,11
30
58. |2x−8|=|x+3|
60. |x−15|=|x+8|
x−15 = x+8 or x −15 = −(x+8)
248 Chapter 9: More on Inequalities
62. |5p+7|=|4p+3|
64. |m−7|=|7−m|
m−7=7−morm−7=−(7 −m)
66. |8−q|=|q+19|
5=7+3x
2or 6−8x
5=−7+3x
2
2−2
3x=4+7
8xor2−2
3x=−4+7
8x
72. |x|≤5
74. |y|>12
76. |x+4|≤9
x<−1or x > 5
{x|x<−1or x > 5},or(−∞,−1) ∪(5,∞)
−1≤x≤1
5
84. |9y−2|≥17
86. |p−2|<6
88. |5x+2|≤13
−3≤x≤11
5
90. |7−2y|>5
Exercise Set 9.3 249
92. |2−9p|≥17
p≤−
3or p ≥19
9,or
−∞,−5
94. |−5−7x|≤30
25
7≥x≥−5
96.
4y−6
>24
100.
1+3x
5
>7
8
3x<−43
102. |t−7|+3≥4
104. 16 ≤|2x−3|+9
2x≤−4or 2x≥10
106.
3x−2
3x≤−3or 3x≥7
=t·5·5
3·5·t
116. l≥w+3,
2
3
2
1
250 Chapter 9: More on Inequalities
118. 1−
1
=3
120. |x−1|=x−1 only when x−1≥0, or x≥1. The solution
124. |y|≤5
Exercise Set 9.4
2. 4x+3y≥0
4. 6y−x>2
6(−2) −5?2
6. Graph: y<3x
10. Graph: y≥x+4
12. Graph: x−y≥3
Graph x−y= 3, using a solid line. Test (0,0): 0 ≥3is
false.
⫺2
4
⫺3
⫺1
5
x
Exercise Set 9.4 251
14. Graph: 2x+3y<6
16. Graph: 2y−x≤4
Graph 2x−4y= 8, using a solid line. Test (0,0): 0 ≥8is
20. Graph: y≥−2
22. Graph: x≤−4
24. Graph: 7x+2y≥21
26. From the graph we see that the related equation is x=1
28. From the graph we see that the related equation is y=−3
30. The equation 4y=8−3xcorresponds to the graph of the
32. Graph: y≥x,
where they overlap.
⫺8
⫺10
34. Graph: y≤x,
y≥−x+3
36. Graph: x≥−2,
x=−2,
38. Graph: x+y≤3,
To find the vertex we solve
x+y=3,
40. Graph: x−y≤2,(1)
42. Graph: y−x≥1,(1)
Think of (3) as two inequalities: