4-8 Multistep Inequalities with Grouping Symbols
Name
Date
Solve. Graph and check the solution.
4(x 3) 16
Inequality with grouping symbols
2
4( x + 3) • ( −2 ) 16 • (2)
Multiply both sides by 2 and
−2
reverse the inequality symbol.
4(x 3) 32
4x 12 32
Apply the Distributive Property.
4x 12 12 32 12
4x 44
Subtract 12 from both sides.
Simplify.
4x 44
4
4
x 11
Divide both sides by 4.
solution
Graph and check:
121110 9 8 7 6 5
Try x 3.
?
4(3 3) 16
2
?
4(0) 16
2
0 16 True
So the solution of 4(x 3) 16 is x 11.
2
Solve each inequality. Graph the solution. Check students’ graphs. Possible graphs shown.
7y 6
1. 3(a 7) 42
2. 20(3 k) 100
3.
20
6
a . 21
Copyright © by William H. Sadlier, Inc. All rights reserved.
0
10
20
30
40
7y ⴙ 6
• 6 . ⴚ20 • 6
6
7y ⴙ 6 . ⴚ120; 7y ⴙ 6 ⴚ 6 . ⴚ120 ⴚ 6
7y ⴚ126
.
; y . ⴚ18
7y . ⴚ126;
7
7
100
ⴚ20(3 ⴙ k)
$
ⴚ20
ⴚ20
3 ⴙ k ≥ ⴚ5
3 ⴙ k ⴚ 3 ≥ ⴚ5 ⴚ 3
3(a ⴚ 7) 42
. 3
3
a ⴚ 7 . 14
a ⴚ 7 ⴙ 7 . 14 ⴙ 7
k $ ⴚ8
50
4. 10 (d 5)
6
ⴚ10 ⴚ8 ⴚ6 ⴚ4 ⴚ2
0
5. 3(2z 6 z) 63
ⴚ(d ⴚ 5)
•6
6
ⴚ60 . ⴚ(d ⴚ 5); ⴚ60 . ⴚd ⴙ 5
ⴚ60 ⴚ 5 . ⴚd ⴙ 5 – 5; ⴚ65 . ⴚd
ⴚ65
ⴚd
,
; d . 65
ⴚ1
ⴚ1
ⴚ20 ⴚ15 ⴚ10 ⴚ5
0
5
6. (4m 12 3m)7 140
6 • (ⴚ10) .
50 60 70 80 90 100
7. 3b 2 5 15
4
b , 14
0
7
14
z.9
0
3
6
9 12 15 18
8. 4n 7 5n 11
2
3b ⴚ 2
ⴙ 5 ⴚ 5 , 15 ⴚ 5
4
3b ⴚ 2
3b ⴚ 2
, 10;
• 4 , 10 • 4
4
4
3b ⴚ 2 , 40; 3b ⴚ 2 ⴙ 2 , 40 ⴙ 2
3b , 42
ⴚ14 ⴚ7
6z ⴚ 18 ⴙ 3z . 63
9z ⴚ 18 ⴙ 18 . 63 ⴙ 18
9z . 81
4n ⴙ 7 ⴚ 5n
• 2 $ 11 • 2
2
4n ⴙ 7 ⴚ 5n $ 22; ⴚn ⴙ 7 $ 22
ⴚn ⴙ 7 ⴚ 7 $ 22 ⴚ 7; ⴚn $ 15
n # ⴚ15
235 230 225 220 215 210
28m ⴚ 84 ⴚ 21m , 140
7m ⴚ 84 ⴙ 84 , 140 ⴙ 84
7m , 224
m , 32
0
8
16
24 32 40
9. 10(c 3) 4c 90
ⴚ10c ⴚ 30 ⴙ 4c $ 90
ⴚ6c ⴚ 30 ⴙ 30 $ 90 ⴙ 30
ⴚ6c 120
#
ⴚ6c $ 120;
ⴚ6
ⴚ6
c # ⴚ20
250 240 230 220 210
0
21
Lesson 4-8, pages 110–111.
Chapter 4
121
For More Practice Go To:
Solve each inequality. Check to justify your work.
11. (p 34) • 7 49
(ⴚp ⴙ 34) • 7
49
.
7
7
ⴚp ⴙ 34 . 7; ⴚp ⴙ 34 ⴚ 34 . 7 ⴚ 34
ⴚp ⴚ27
ⴚp . ⴚ27;
,
; p , 27
ⴚ1
ⴚ1
7(t ⴚ 7) 49
, 7
7
t27,7
t2717,717
t , 14
?
?
Check: 7(10 2 7) ⬍ 49
Check: (ⴚ20 ⴙ 34) • 7 ⬎ 49
?
7(3) ⬍ 49
21 , 49 True
?
(14)7 ⬎ 49
98 . 49 True
13. 7(3 s) (10s) 147 14. 3d 2d 3 18
4
ⴚ21 ⴚ 7s ⴙ 10s . ⴚ147
ⴚ21 ⴙ 3s . ⴚ147
ⴚ21 ⴙ 3s ⴙ 21 . ⴚ147 ⴙ 21
3s . ⴚ126
s . ⴚ42
?
Check: ⴚ7(3 ⴙ 0) ⴚ (ⴚ10 • 0) ⬎ ⴚ147
ⴚ21 . ⴚ147 True
16. 62 24(2) 5x 4x
2
9x
; 168 # 9x
2
x $ 1823
? 5(20) ⴙ 4(20)
Check: 62 ⴙ 24(2) ⱕ
2
84 # 90 True
36 ⴙ 48 #
22
1
ⴚ6c ⴚ 24 1 3c ,
2
1
1
ⴚ3c , 24 ; c . ⴚ8
2
6
? 2(4 ⴚ 3)
Check: ⴚ6(0 1 4) 1 3(0) ⬍
2ⴙ2
1
ⴚ24 , True
2
19. A cab ride costs $5 for the first mile and $3
for each additional mile. If a cab ride costs no
more than $23, how many miles was the ride?
Let m ⴝ number of miles.
5 ⴙ 3(m ⴚ 1) # 23; 5 ⴙ 3(m ⴚ 1) ⴚ 5 # 23 ⴚ 5
3(m ⴚ 1) # 18; 3m ⴚ 3 # 18; 3m ⴚ 3 ⴙ 3 # 18 ⴙ 3
3m # 21; m # 7; The ride was no more than 7 miles.
x ⴙ 4 # ⴚ100
x ⴙ 4 ⴚ 4 # ⴚ100 ⴚ 4
x # ⴚ104
ⴚ109 ⴙ 4 ?
Check:
ⱖ 20
ⴚ5
ⴚ105 ?
ⱖ 20; 21 $ 20 True
ⴚ5
15. 2r 5 r 4 3 6
2
5d ⴚ 3
• 4 . 18 • 4
4
5d ⴚ 3 . 72; 5d ⴚ 3 + 3 . 72 + 3
5d . 75; d . 15
3(21) ⴙ 2(21) ⴚ 3 ?
Check:
⬎ 18
4
102 ?
63 ⴙ 42 ⴚ 3 ?
⬎ 18;
⬎ 18
4
4
1
25 . 18 True
2
2(4 3)
17. 6(c 4) 3c 12. x 4 20
5
xⴙ4
• (ⴚ5) # 20 • (ⴚ5)
ⴚ5
3r ⴙ 9
ⴙ3ⴚ3.6ⴚ3
2
3r ⴙ 9
3r ⴙ 9
•2.3•2
. 3;
2
2
3r ⴙ 9 . 6; 3r ⴙ 9 ⴚ 9 . 6 ⴚ 9
3r . ⴚ3; r . ⴚ1
?
2(0) ⴙ 5 ⴙ 0 ⴙ 4
Check:
ⴙ3⬎6
2
?
9
1
ⴙ 3 ⬎ 6; 7 . 6 True
2
2 2
16t
2
18. 5 • 2 7 2t(2 9) 1
16t
; 228 . 108t; t , 2
9
4
?
(1)
16
Check: 52 • 2 1 7 ⬎ 2(1)(22 ⴙ 9) ⴙ
4
57 . 26t 1
57 . 30 True
20. A rectangular plot of land has a perimeter
that does not exceed 36.4 meters. If the
length of the plot is 3 times the width, what
is the length of the rectangular plot?
Let w ⴝ width of plot; let ᐉ ⴝ length of plot.
2(4w) 36.4
#
2(w ⴙ 3w) # 36.4;
2
2
4w
18.2
4w # 18.2;
#
4
4
w # 4.55; ᐉ ⴝ 3w ⴝ 3(4.55) ⴝ 13.65
The length of the plot does not exceed 13.65 meters.
21. Sarah wants to earn at least $200 for her charity while running a race. She
will earn $25 plus $8 for each mile she runs. If x represents the number
of miles she runs, which inequality represents the amount of money Sarah
wants to earn?
A. $8x $25 $200
122
Chapter 4
B. $8x $25 $200
4
C. $25x $8 $200
D. $25x $8 $200
Copyright © by William H. Sadlier, Inc. All rights reserved.
10. 7(t 7) 49