Chapter 7 Lesson 5 Solving Inequalities by Multiplying or Dividing pgs. 350 - 354

1. Chapter 7 Lesson 5Solving Inequalities by Multiplying or Dividingpgs. 350 - 354 What you’ll learn: Solve inequalities by multiplying or dividing by positive & negative numbers

2. 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

3. Key Concept Continued: • Examples: 2 < 6 3 > -9 4(2) < 4(6) 3 > -9 8 < 24 3 3 1 > -3 These properties are also true for a  b and a  b

4. Example 1: Multiply or Divide by a Positive Number Write the inequality: 7y > 63 Divide each side by 7: 7y > 63 7 7 Simplify: y > 9 The solution is y > 9. You can check this solution by substituting a number greater than 9 into the inequaltiy. • Solve 7y > 63 Check your solution Check: Let’s check with 11 7(11) > 63 77 > 63 

5. Example 1: Another Look 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. • Solve 6 x Check your solution 7 Check using 35: 6 35 6  5  7

6. 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. 20 Solve: 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.

7. What happens when each side of an inequality is multiplied or divided by a negative number? -6 < 11 Multiply each side by -1: -1(-6) < -1(11) This inequality is false: 6 < -11 10  5 Divide each side by -5: 10  5 -5 -5 This 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

8. 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

9. 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 

10. Example 3: Divide by a Negative Number • Solve each inequality and check your solution. Then graph the solution on a number line. 15  -5b Divide each side by -5 and reverse the symbol: 15 -5b -5 -5 Check this result: -3  b or b  -3 You 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.

11. Example 3: Multiply by a Negative Number • Solve the inequality, check your solution and graph the solution on a number line. 6 > x -7 Multiply 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.

12. Your Turn!!Solve, check and graph each inequality (-3) s -3.5(-3) 3 s  10.5 • s -3.5 3 • 15 > 3t • 13a  -26 • 7  h -14 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

13. Extra Practice Is By The Door On Your Way Out! • Don’t Let The Negative Signs Trip You Up!!