Chapter 7 Lesson 5Solving Inequalities by Multiplying or Dividing

pgs. 350 - 354What you’ll learn:

Solve inequalities by multiplying or dividing by positive & negative

numbers

Key Concept: Multiplication & Division

Properties (pg. 350)» Words: When you multiply or divide

each side of an inequality by the same 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 > b c c

2. If a < b, then ac<bc and a < b c c

Key Concept Continued:

» Examples:

2 < 6 3 > -94(2) < 4(6) 3 > -9

8 < 24 3 31 > -3

These properties are also true for a b and a b

Example 1: Multiply or Divide by a Positive Number

» Solve 7y > 63 Check your solution

Write the inequality: 7y > 63 Divide each side by 7: 7y > 63

7 7Simplify: y > 9

The solution is y > 9. You can check this solution by substituting a number greater than 9 into the inequaltiy.

Check: Let’s check with 11 7(11) > 63

77 > 63

Example 1: Another Look

» Solve 6 x Check your solution 7

Write the inequality: 6 x 7

Multiply each side by 7: (7)6 x(7) 7

Simplify: 42 x which also means x 42

The solution is x 42 You can check this solution by substituting 42 or a number less than 42 into the inequality.

Check using 35: 6 35 6 5 7

Example 2: Write an inequality

Julia delivers pizza on weekends. Her average tip is $1.50 for each pizza that she delivers. How many pizzas must she deliver to earn at least $20 in tips?

A. 10 B. 13 C. 14 D. 20Solve: Let x represent the number of pizzas.

1.50 = average per pizza = times x = number of pizzas = at least 20 = total amount to earn

1.50x 20

This works out to 13.333,So at least 14 pizzas.

What happens when each side of an inequality is multiplied or divided by

a negative number? -6 < 11Multiply each side by -1: -1(-6) < -1(11) This inequality is false: 6 < -11

10 5Divide each side by -5: 10 5

-5 -5This inequality is false: -2 -1

The inequalities 6 < -11 and -2 > -1 are both false. However,They would both be true if the inequality symbols were reversed.Change < to > and change > to <. 6 > -11 TRUE -2 < -1 TRUE

Key Concept: Multiplication & Division

Properties (352)Words: When you multiply or divide each of an inequality by the

same negative number, the inequality symbol must be REVERSED for the inequality to remain true.

Symbols: For all numbers a, b, c, where c 0,1. If a > b, then ac < bc and a < b

c c

2. If a < b, then ac> bc and a > b c c

Key Concept Continued:

» Examples:

7 > 1 -4 < 16-2(7) < -2(1) Reverse the symbols-4 16 -14 < -2 -4 -4

1 > -4

This is also true when using and

Example 3: Divide by a Negative Number

» Solve each inequality and check your solution. Then graph the solution on a number line.

15 -5bDivide each side by -5 and reverse the symbol: 15 -5b -5 -5Check this result: -3 b or b -3You can check this result by replacing x in the

original equation with -3 or a number less than -3

Check using -4:15 -5(-4)15 20

See the board for the graph.

Example 3: Multiply by a Negative Number

» Solve the inequality, check your solution and graph the solution on a number line.

6 > x

-7Multiply each side by -7 and reverse the

symbol: -7(6) < x (-7) -7

Check this result: -42 x or x > -42

Check by putting a number greater than -42 in the original inequality. Check using -35:

6 > -35 = 6 > 5 -7

See graph on board.

Your Turn!!Solve, check and graph each

inequality

A. s -3.5 3

B. 15 > 3t

C. 13a -26

D. 7 h -14

(-3) s -3.5(-3) 3 s 10.5

15 3t 3 3 5 > t or t 5

13a -26 13 13 a -2

(-14)7 h (-14) -14

-98 h or h -98

» Extra Practice Is By The Door On Your Way Out!

» Don’t Let The Negative Signs Trip You Up!!