LAB: Inequalities with Negative Coefficients p.304 Q U E ST ION: How do you solve an inequality with a negative coefficient?

5.2 Solve Inequalities Using Multiplication and Division You will solve inequalities using multiplication and division. Essential Question: How do you solve inequalities using multiplication and division? You will learn how to answer this question by applying multiplication and division properties of inequality and by learning when to reverse the direction of the inequality symbol.

Warm-Up Exercises Solve an inequality using multiplication EXAMPLE 1 x 4 < 5 Write original inequality. x 4 < Multiply each side by 4. x < 20 Simplify. ANSWER The solutions are all real numbers less than 20. Check by substituting a number less than 20 in the original inequality.. Graph your solution. x 4 < 5 Solve

Warm-Up Exercises GUIDED PRACTICE for Example 1 Solve the inequality. Graph your solution. 1. > 8 x 3 x > 24 ANSWER

Warm-Up Exercises GUIDED PRACTICE for Example 1 2.  –2 m 8 ANSWER m  –16 Solve the inequality. Graph your solution.

Warm-Up Exercises GUIDED PRACTICE for Example 1 ANSWER y  –10 3.  –4 y 2.5 Solve the inequality. Graph your solution.

Warm-Up Exercises Solve an inequality using multiplication EXAMPLE 2 < 7< 7 x –6 Write original inequality. Multiply each side by –6. Reverse inequality symbol. x > –42 Simplify. ANSWER The solutions are all real numbers greater than –42. Check by substituting a number greater than –42 in the original inequality. < 7. Graph your solution. x –6 Solve x –6 > –6 –6 7

Warm-Up Exercises Solve an inequality using multiplication EXAMPLE 2 CHECK Write original inequality. Substitute 0 for x. Solution checks x –6 < 7 0 –6 < 7 ? 0 < 7

Warm-Up Exercises Solve an inequality using division EXAMPLE 3 –3x > 24 –3x –3 < 24 –3 x < –8 Write original inequality. Divide each side by –3. Reverse inequality symbol. Simplify. –3x > 24. Solve If you graph x < 28, would you use an open or a closed circle?

Warm-Up Exercises GUIDED PRACTICE for Examples 2 and 3 4. > 12 x –4 x < –48 ANSWER Solve the inequality. Graph your solution.

Warm-Up Exercises GUIDED PRACTICE for Examples 2 and 3 5. < 1.6 m –7 m > –11.2 ANSWER Solve the inequality. Graph your solution.

Warm-Up Exercises GUIDED PRACTICE for Examples 2 and v  45 v  9 ANSWER Solve the inequality. Graph your solution.

Warm-Up Exercises GUIDED PRACTICE for Examples 2 and 3 7. –6n < 24 n > –4 ANSWER Solve the inequality. Graph your solution.

Warm-Up Exercises SOLUTION Standardized Test Practice EXAMPLE 4 A student pilot plans to spend 80 hours on flight training to earn a private license. The student has saved $6000 for training. Which inequality can you use to find the possible hourly rates r that the student can afford to pay for training ? 80r  6000 A 80r  6000 B 6000r  80 C 6000r  80 D The total cost of training can be at most the amount of money that the student has saved. Write a verbal model for the situation. Then write an inequality.

Warm-Up Exercises Standardized Test Practice EXAMPLE 4 ANSWER The correct answer is B. A DCB 80 r < – 6000

Warm-Up Exercises Solve a real-world problem EXAMPLE 5 PILOTING In Example 4, what are the possible hourly rates that the student can afford to pay for training ?

Warm-Up Exercises Solve a real-world problem EXAMPLE 5 SOLUTION Write inequality. 80r 80 ≤ Divide each side by 80. r ≤ 75 Simplify. ANSWER The student can afford to pay at most $75 per hour for training. 80 r ≤ 6000

Warm-Up Exercises GUIDED PRACTICE for Examples 4 and 5 8. WHAT IF ? In Example 5, suppose the student plans to spend 90 hours on flight training and has saved $6300. Write and solve an inequality to find the possible hourly rates that the student can afford to pay for training. 90r  6300, r  $70/h ANSWER

You will solve inequalities using multiplication and division. Essential Question: How do you solve inequalities using multiplication and division? The multiplication and division properties of inequality can be used to produce equivalent inequalities. Reverse the direction of the inequality symbol when multiplying or dividing by a negative number. Multiply or divide each side of the inequality by the same number to isolate the variable and produce an equivalent inequality. Reverse the direction of the inequality symbol if you multiply or divide by a negative number.

Warm-Up Exercises Daily Homework Quiz Solve the inequality. Graph your solution. 1. –72 < 8p ANSWER p > –9 2. w – 6 – > –5 ANSWER w  30

Warm-Up Exercises Daily Homework Quiz A fitness walking group walks 4 days each week with a goal of at least 12 miles per week. Write and solve an inequality to find the average number of possible miles m the group should walk each day to meet its goal. 3. ANSWER 4m  12; an average of at least 3 miles per day.