8-8 multiplying or dividing - glencoe/mcgraw-hill · · 2009-06-08multiplying 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.

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

Gr8 MS Math SE ©09 - 874050449_453_C08_L08_874050.indd 449449_453_C08_L08_874050.indd 449 09/22/07 12:46:54 PM09/22/07 12:46:54 PM

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



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


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



8-8 multiplying or dividing - glencoe/mcgraw-hill · · 2009-06-08multiplying or dividing ......

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