8-8
MAIN IDEA
Solve inequalities by
using the Multiplication
or Division Properties of
Inequality.
Math Online
glencoe.com
Solving Inequalities by
Multiplying or Dividing
COINS Lamar, Mario, and Nick put the
money from their pockets on the table.
Lamar had more money than Nick
since $1.70 > $1.40. Will this still be
true if each boy donates half their
money to the school fundraiser?
1. Divide each side of the inequality
1.70 > 1.40 by 2. Write the resulting
inequality and decide if it is true or
false.
• Extra Examples
• Personal Tutor
• Self-Check Quiz
Name
Amount of Money
Lamar
1 dollar bill, 2 quarters,
2 dimes
Mario
1 dollar bill, 3 quarters,
1 dime, 1 nickel
Nick
5 quarters, 1 dime,
1 nickel
2. Who would have more if Mario and Lamar tripled their money by
doing lawn work at home? Explain.
The examples above demonstrate additional properties of inequality.
These properties are also true for a ≥ b and a ≤ b.
Properties of Inequality
Key Concept
Words
When you multiply or divide each side of an inequality by a
positive number, the inequality remains true.
Symbols
For all numbers a, b, and c, where c > 0,
_
1. if a > b, then ac > bc and _
c > c.
a
b
_
2. if a < b, then ac < bc and _
c < c.
a
Examples
b
5<8
2 > -10
4(5) < 4(8)
-10
2
_
>_
2
2
20 < 32
1 > -5
Solve Inequalities by Dividing
Checking Solutions
You can check the solution
in Example 1 by substituting
numbers greater than -6
into the inequality and
testing it to verify that it
holds true.
1 Solve 7y > -42. Check your solution.
7y > -42
Write the inequality.
7y
-42
_
>_
Divide each side by 7.
7
7
y > -6
Simplify.
The solution is y > -6.
Lesson 8-8 Solving Inequalities by Multiplying or Dividing
449
Solve Inequalities by Multiplying
1
2 Solve _
x ≤ 8. Check your solution.
3
_1 x ≤ 8
3
Write the inequality.
1
3 _
x ≤ 3(8) Multiply each side by 3.
(3 )
x ≤ 24
Simplify.
The solution is x ≤ 24. You can check this solution by substituting 24
and a number less than 24 into the inequality.
Solve each inequality. Check your solution.
a. 3a ≥ 45
b.
n
_
< -16
c. 81 ≤ 9p
4
What happens when each side of an inequality is multiplied or divided
by a negative number?
Graph 3 and 5 on a number line.
Multiply each number by -1.
-5-4-3-2 -1 0 1 2 3 4 5
-5-4-3-2 -1 0 1 2 3 4 5
Since 3 is to the left of 5, 3 < 5.
Since -3 is to the right of -5,
-3 > -5.
Notice that the numbers being compared switched positions as a result
of being multiplied by a negative number. In other words, their order
reversed.
These and other examples suggest the following properties. These
properties also hold true for a ≥ b and a ≤ b.
Properties of Inequality
Common Error
Do not reverse the
inequality symbol just
because there is a
negative sign in the
inequality, as in 7y < -42.
Only reverse the inequality
symbol when you multiply
or divide each side by a
negative number.
Words
When you multiply or divide each side of an inequality by a
negative number, the direction of the inequality symbol must
be reversed for the inequality to remain true.
Symbols
For all numbers a, b, and c, where c < 0,
_
1. if a > b, then ac < bc and _
c < c.
a
b
_
2. if a < b, then ac > bc and _
c > c.
a
Examples
b
8>5
-1(8) < -1(5)
-8 < -5
450
Key Concept
Chapter 8 Algebra: More Equations and Inequalities
-3 < 9
Reverse the inequality symbols.
-3
9
_
>_
-3
-3
1 > -3
Multiply or Divide by a Negative Number
_
3 Solve a ≥ 8. Check your solution.
-2
a
_
≥8
-2
a
-2 _
≤ -2(8)
-2
( )
a ≤ -16
Write the inequality.
Multiply each side by -2 and reverse the inequality symbol.
Check this result.
4 Solve -24 > -6n. Check your solution.
-24 > -6n
Write the inequality.
-6n
-24
_
<_
Divide each side by -6 and reverse the symbol.
-6
-6
4 < n or n > 4
d.
Check this result.
c
_
< -14
-7
e. -5d ≥ 30
f. -3 ≤ _
w
-8
Some inequalities involve more than one operation. To solve, work
backward as you did in solving two-step equations.
5 BASEBALL Manny was trying to break his school’s record by getting
61 hits in one season. Halfway through the season he already had
34 hits. Manny averages 2 hits per game. Write and solve an
inequality to find how many more games it will take at that rate for
Manny to have at least 61 hits. Interpret the solution.
The phrase at least means greater than or equal to. Let g = the number
of games he needs to play. Then write an inequality.
34 + 2g ≥ 61
34 - 34 + 2g ≥ 61 - 34
Write the inequality.
Subtract 34 from each side.
2g ≥ 27
Simplify.
2g
27
_
≥_
2
2
Divide each side by 2.
g ≥ 13.5
Simplify.
If Manny plays only entire games, he should have 61 hits after 14
more games. Manny should break the record.
g. DVDS Joan has a total of $250. DVDs cost $18.95 each. Write and
solve an inequality to find how many DVDs she can buy and still
have at least $50. Interpret the solution.
Examples 1–2
(pp. 449–450)
Examples 3–4
Solve each inequality. Check your solution.
1. 3x > 12
5. -4y > 32
2.
_3 < _7 y
4
3. 8x ≤ -72
9
6. -56 ≤ -7p
7.
(p. 451)
Example 5
(p. 451)
HOMEWORK
HELP
For
Exercises
See
Examples
10–15
1, 2
16–27
28–29
3, 4
5
g
_
< -7
-2
4.
_h ≥ -6
8.
d
_
≥ -3
4
-3
9. RENTAL CARS A rental car company charges $45 plus an additional $0.19 per
mile to rent a car. If Lawrence does not want to spend more than $100 for
his rental car, write and solve an inequality to find how many miles he can
drive and not spend more than $100. Interpret the solution.
Solve each inequality. Check your solution.
10. 5x < 15
11. 9n ≤ 45
12. 14k ≥ -84
13. -12 > 3g
14. -100 ≤ 50p
15. 2y < -22
16. -4w ≥ 20
17. -3r > 9
18. -72 < -12h
19. -6c ≥ -6
20.
x
22. _ ≤ -3
9
t
25. _ ≤ -2
-5
v
_
>4
-4
n
23. _ < -14
7
y
26. -8 ≤ _
0.2
21.
a
_
≥5
-3
m
24. _ < -7
-2
-1
27. _k > -10
2
28. GYM MEMBERSHIP A local gym charges $5 each time you enter. They also
have yearly memberships for $190. Write and solve an inequality to find
how many times a person should use the gym so that a yearly membership
is less expensive than paying each time. Interpret the solution.
29. WORK Max charges $6.25 an hour to rake leaves. He is trying to save
enough money for a new pair of shoes that cost $89. Write and solve an
inequality to find how many whole hours Max must work to buy the shoes.
Interpret the solution.
Solve each inequality. Graph the solution set on a number line.
30. 6y > 15 + y
31. 8k + 3 ≤ -5
33. 7 + _ < 4
34.
n
3
w
_
- 4 ≤ -5
8
32. -5g + 5 ≥ -7 - 2g
35. 10 - 3x ≥ 25 + 2x
Write an inequality for each sentence. Then solve the inequality.
36. Three times a number increased by four is less than -62.
37. The quotient of a number and -5 increased by one is at most 7.
EXTRA PRACTICE
See pages 691, 707.
452
38. The quotient of a number and 3 minus two is at least -12.
39. The product of -2 and a number minus six is greater than -18.
Chapter 8 Algebra: More Equations and Inequalities
H.O.T. Problems
40. OPEN ENDED Write an inequality for the following sentence and then solve.
The quotient of a number and -6 increased by 5 is at most 9.
Name three numbers that are possible solutions for x. Explain.
41. FIND THE ERROR Sonia and Kendra each solved 7x ≤ -49. Who is correct?
Explain.
Sonia
7x ≤ -49
Kendra
7x ≤ -49
7x
-49
_
≥_
7x
-49
_
≤_
x ≥ -7
x ≤ -7
7
7
7
7
42. CHALLENGE In five games, you score 16, 12, 15, 13, and 17 points. How
many points must you score in the sixth game to have an average of at
least 15 points?
43.
WR ITING IN MATH Explain when you should reverse the inequality
symbol when solving an inequality.
44. Which is a possible value of x if the
45. You want to purchase a necklace for
area of the trapezoid is less than 256
square feet?
$325. You have already saved $115 and
can set aside $22 a week. Which
inequality can be used to find the
number of weeks it will take to save at
least $325?
16.5 ft
x ft
F 22w + 115 ≥ 325
G 22w + 115 ≤ 325
20 ft
H 22 + 115w ≤ 325
A 14
C 16
B 15
D 17
J
22w + 115 < 325
Solve each inequality. Check your solution.
(Lesson 8-7)
46. y + 7 < 9
48. j - 8 ≥ -12
47. a - 5 ≤ 2
Write an inequality for each sentence.
49. -14 > 8 + n
(Lesson 8-6)
50. SPEED A minimum speed on a certain highway is 45 miles per hour.
51. BIRDS A hummingbird’s wings can beat up to 200 times per second.
52. MEASUREMENT Three boxes with height 12 inches, width 10 inches,
and length 13 inches are stacked on top of each other. What is the
volume of the space that they occupy? (Lesson 7-5)
Lesson 8-8 Solving Inequalities by Multiplying or Dividing
453