5.3 Solve Multi-Step Inequalities You will solve multi- step inequalities. Essential question: How do you solve multi-step inequalities? You will learn how to answer this question by using the distributive property and the properties of inequality.

Warm-Up Exercises Solve a two-step inequality EXAMPLE 1 3x – 7 < 8 Write original inequality. 3x < 15 Add 7 to each side. x < 5 Divide each side by 3. ANSWER The solutions are all real numbers less than 5. Check by substituting a number less than 5 in the original inequality. 3x – 7 < 8. Graph your solution. Solve

Warm-Up Exercises Solve a two-step inequality EXAMPLE 1 CHECK Write original inequality. Solution checks. Substitute 0 for x. ? 3(0) – 7 < 8 3x –7 < 8 –7 < 8

Warm-Up Exercises Solve a multi-step inequality EXAMPLE 2 Write original inequality. –0.6 ( x – 5)  15 –0.6x + 3  15 Distributive property Subtract 3 from each side. x  –20 Divide each side by 0.6. Reverse inequality symbol. – Solve –0.6(x – 5)  15. –0.6x  12 Is -21 a solution of the inequality? Explain.

Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 Solve the inequality. Graph your solution. 2x – < – x 14 < – ANSWER

Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 Solve the inequality. Graph your solution. –6y + 5 – < – y 3.5 > – ANSWER

Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 3. –1 4 ( p –12) > –2 p < 20 ANSWER Solve the inequality. Graph your solution.

Warm-Up Exercises Solve a multi-step inequality EXAMPLE 3 6x – 7 > 2x + 17 Write original inequality. 6x > 2x + 24 Add 7 to each side. 4x > 24 Subtract 2x from each side. x > 6 Divide each side by 4. ANSWER The solutions are all real numbers greater than 6. 6x – 7 > 2x Graph your solution. Solve

Warm-Up Exercises Identify the number of solutions of an inequality EXAMPLE 4 Solve the inequality, if possible. a. 14x + 5 < 7(2x – 3) SOLUTION 14x + 5 < 7(2x – 3) a.a. Write original inequality. 14x + 5 < 14x – 21 Distributive property 5 < –21 Subtract 14x from each side. ANSWER There are no solutions because 5 < –21 is false.

Warm-Up Exercises Identify the number of solutions of an inequality EXAMPLE 4 12x – 1 > 6(2x – 1) Write original inequality. Distributive property Subtract 12x from each side. 12x – 1 > 12x – 6 –1 > –6 ANSWER All real numbers are solutions because –1 > –6 is true. b. 12x – 1 > 6(2x – 1)

Warm-Up Exercises Solve the inequality, if possible. Graph your solution. GUIDED PRACTICE for Examples 3 and 4 5x – 12  3x – 4 4. x  4 ANSWER

Warm-Up Exercises Identify the number of solutions of an inequality EXAMPLE (m + 5) < 5m + 17 GUIDED PRACTICE for Examples 3 and 4 ANSWER no solution Solve the inequality, if possible. Graph your solution.

Warm-Up Exercises Identify the number of solutions of an inequality EXAMPLE 4 GUIDED PRACTICE for Examples 3 and 4 1 – 8s –4(2s – 1) 6. < – Solve the inequality, if possible. Graph your solution. all real numbersANSWER

Warm-Up Exercises Use the sign shown. A gas station charges $.10 less per gallon of gasoline if a customer also gets a car wash. What are the possible amounts (in gallons) of gasoline that you can buy if you also get a car wash and can spend at most $20? Car Wash EXAMPLE 5 Solve a multi-step problem

Warm-Up Exercises SOLUTION EXAMPLE 5 Solve a multi-step problem STEP 1 Write a verbal model. Then write an inequality. Because you are getting a car wash, you will pay $2.09 – $.10 = $1.99 per gallon of gasoline. Let g be the amount (in gallons) of gasoline that you buy g 820 < – Price of gasoline (dollars/gallon) Amount of gasoline (gallons) Price of car wash (dollars) Maximum amount (dollars) + <

Warm-Up Exercises EXAMPLE 5 Solve a multi-step problem STEP 2 Solve the inequality. Write inequality. Subtract 8 from each side. 1.99g ≤ g + 8 ≤ 20 Divide each side by g ≤ ANSWER You can buy up to slightly more than 6 gallons of gasoline.

Warm-Up Exercises CHECK You can use a table to check the reasonableness of your answer. The table shows that you will pay $19.94 for exactly 6 gallons of gasoline. Because $19.94 is less than $20, it is reasonable to conclude that you can buy slightly more than 6 gallons of gasoline. EXAMPLE 5 Solve a multi-step problem Gasoline (goal) Total amount spent (dollars)

Warm-Up Exercises GUIDED PRACTICE for Example 3 and 4 GUIDED PRACTICE for Example 5 WHAT IF? In Example 5, suppose that a car wash costs $9 and gasoline regularly costs $2.19 per gallon. What are the possible amounts (in gallons) of gasoline that you can buy ? 7. g ≤ 5.02 gal ANSWER

Warm-Up Exercises EXAMPLE 5 Solve a multi-step problem GUIDED PRACTICE for Example 5 8. You are saving money for a summer camp that costs $1800. You have saved $500 so far, and you have 14 more weeks to save the total amount. What are the possible average amounts of money that you can save per week in order to have a total of at least $1800 saved? Camp Costs at least $ 92.86/wk ANSWER

You will solve multi- step inequalities. Essential question: How do you solve multi-step inequalities? You can solve multi-step inequalities by applying the same steps used for solving multi-step equations, with the additional step of reversing an inequality symbol if you multiply or divide by a negative number. Use the distributive property and the properties of inequality to produce equivalent inequalities. Reverse the direction of the inequality symbol when dividing or multiplying by a negative number. If you reach an inequality that is false, there are no solutions; if you reach an inequality that is always true, then all real numbers are solutions.

Warm-Up Exercises Daily Homework Quiz 1. Solve 3(x + 8) < 9. Graph your solution. ANSWERall real numbers less than –5

Warm-Up Exercises Daily Homework Quiz 3. 5(a – 2)< 5a – 14 ANSWER no solutions because –10 < –14 is false Solve the inequality, if possible. 2. 6k + 1 > 3(2k – 3) ANSWER all real numbers because 1 > –9 is true

Warm-Up Exercises Daily Homework Quiz 4. A box of cat treats contains at least 50 treats. So far you have fed your cats 18 treats. If you want the box of treats to last 8 days, what are the possible average numbers of treats you can feed the cats per day? ANSWER an average of no more than 4 treats per day.