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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.
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.
Solve Inequalities by Dividing
1 Solve 7y > -42. Check your solution.
7y > -42 Write the inequality.7y
_ 7 >
-42 _
7 Divide each side by 7.
y > -6 Simplify.
The solution is y > -6.
Solving Inequalities by Multiplying or Dividing
Checking SolutionsChecking SolutionsYou 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.
8-8
Name Amount of Money
Lamar1 dollar bill, 2 quarters, 2 dimes
Mario1 dollar bill, 3 quarters, 1 dime, 1 nickel
Nick5 quarters, 1 dime, 1 nickel
MAIN IDEASolve inequalities by using the Multiplication or Division Properties of Inequality.
Math Online
glencoe.com• Extra Examples• Personal Tutor• Self-Check Quiz
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 a _ c > b _ c .
2. if a < b, then ac < bc and a _ c < b _ c .
Examples 5 < 8 2 > -10
4(5) < 4(8) 2 _ 2 > -10 _ 2
20 < 32 1 > -5
Key ConceptProperties of Inequality
Lesson 8-8 Solving Inequalities by Multiplying or Dividing 449
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Solve Inequalities by Multiplying
2 Solve 1 _ 3 x ≤ 8. Check your solution.
1 _ 3 x ≤ 8 Write the inequality.
3 ( 1 _ 3 x) ≤ 3(8) Multiply each side by 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 _ 4 < -16 c. 81 ≤ 9p
What happens when each side of an inequality is multiplied or divided
by a negative number?
Graph 3 and 5 on a number line.
-2-3-4-5 543-1 210
Since 3 is to the left of 5, 3 < 5.
Multiply each number by -1.
-2-3-4-5 543-1 210
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.
Common ErrorCommon ErrorDo 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 a _ c < b _ c .
2. if a < b, then ac > bc and a _ c > b _ c .
Examples 8 > 5 -3 < 9
-1(8) < -1(5) Reverse the inequality symbols. -3 _ -3 > 9 _
-3
-8 < -5 1 > -3
Key ConceptProperties of Inequality
450 Chapter 8 Algebra: More Equations and Inequalities
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Lesson 8-8 Solving Inequalities by Multiplying or Dividing 451
Multiply or Divide by a Negative Number
3 Solve a _ -2
≥ 8. Check your solution.
a _ -2
≥ 8 Write the inequality.
-2 ( a _ -2
) ≤ -2(8) Multiply each side by -2 and reverse the inequality symbol.
a ≤ -16 Check this result.
4 Solve -24 > -6n. Check your solution.
-24 > -6n Write the inequality.
-24 _ -6
< -6n _ -6
Divide each side by -6 and reverse the symbol.
4 < n or n > 4 Check this result.
d. c _ -7
< -14 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 Write the inequality.
34 - 34 + 2g ≥ 61 - 34 Subtract 34 from each side.
2g ≥ 27 Simplify.
2g
_ 2 ≥ 27 _
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.
449_453_C08_L08_874050.indd 451 5/25/09 11:16:20 PM
Solve each inequality. Check your solution. Examples 1–2(pp. 449–450)
1. 3x > 12 2. 3 _ 4 < 7 _
9 y 3. 8x ≤ -72 4. h _
4 ≥ -6
Examples 3–4(p. 451)
5. -4y > 32 6. -56 ≤ -7p 7. g _
-2 < -7 8. d _
-3 ≥ -3
Example 5(p. 451)
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. v _ -4
> 4 21. a _ -3
≥ 5
22. x _ 9 ≤ -3 23. n _
7 < -14 24. m _
-2 < -7
25. t _ -5
≤ -2 26. -8 ≤ y _
0.2 27. -1
_ 2 k > -10
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 32. -5g + 5 ≥ -7 - 2g
33. 7 + n _ 3 < 4 34. w _
8 - 4 ≤ -5 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.
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.
For Exercises
See Examples
10–15 1, 2
16–2728–29
3, 45
HOMEWORK HELPHELP
See pages 691, 707.EXTRA PRACTICEPRACTICE
452 Chapter 8 Algebra: More Equations and Inequalities
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Solve each inequality. Check your solution. (Lesson 8-7)
46. y + 7 < 9 47. a - 5 ≤ 2 48. j - 8 ≥ -12 49. -14 > 8 + n
Write an inequality for each sentence. (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)
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
7x _ 7 ≥ -49 _
7
x ≥ -7
Kendra 7x ≤ -49
7x _ 7 ≤ -49 _
7
x ≤ -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. MATHWRITING IN Explain when you should reverse the inequality
symbol when solving an inequality.
H.O.T. Problems
44. Which is a possible value of x if the
area of the trapezoid is less than 256
square feet?
20 ft
x ft
16.5 ft
A 14 C 16
B 15 D 17
45. You want to purchase a necklace for
$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?
F 22w + 115 ≥ 325
G 22w + 115 ≤ 325
H 22 + 115w ≤ 325
J 22w + 115 < 325
Lesson 8-8 Solving Inequalities by Multiplying or Dividing 453
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