Solve Multi-Step Inequalities Warm Up Lesson Presentation Lesson Quiz

Warm-Up 1. Is 4 a solution of the inequality 2a – 1.5  6? ANSWER yes 2. Solve the equation of + 8 = 7. f 4 ANSWER –4 3. All hats sell for the same price at a swap meet. You buy one pair of sunglasses for $7 and 2 hats. How much did each hat cost? ANSWER $9.75

3x – 7 < 8. Graph your solution. Solve Example 1 3x – 7 < 8. Graph your solution. Solve 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.

Example 1 CHECK 3x –7 < 8 Write original inequality. ? 3(0) – 7 < 8 Substitute 0 for x. –7 < 8 Solution checks.

Example 2 Solve –0.6(x – 5)  15. –0.6(x – 5)  15 –0.6x + 3  15 Write original inequality. –0.6x + 3  15 Distributive property –0.6x  12 Subtract 3 from each side. x  –20 Divide each side by 0.6. Reverse inequality symbol. –

Guided Practice Solve the inequality. Graph your solution. 2x – 5 23. 1. < – x 14 < – ANSWER –6y + 5 –16. 2. < – y 3.5 > – ANSWER 3. –1 4 ( p –12) > –2 p < 20 ANSWER

6x – 7 > 2x + 17. Graph your solution. Solve Example 3 6x – 7 > 2x + 17. Graph your solution. Solve 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.

Solve the inequality, if possible. a. 14x + 5 < 7(2x – 3) Example 4 Solve the inequality, if possible. a. 14x + 5 < 7(2x – 3) SOLUTION 14x + 5 < 7(2x – 3) 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.

All real numbers are solutions because –1 > –6 is true. Example 4 b. 12x – 1 > 6(2x – 1) 12x – 1 > 6(2x – 1) Write original inequality. 12x – 1 > 12x – 6 Distributive property –1 > –6 Subtract 12x from each side. ANSWER All real numbers are solutions because –1 > –6 is true.

Guided Practice Solve the inequality, if possible. Graph your solution. 4. 5x – 12  3x – 4 x  4 ANSWER 5. 5(m + 5) < 5m + 17 ANSWER no solution 1 – 8s –4(2s – 1) 6. < – all real numbers ANSWER

Example 5 Car Wash Use the sign shown. A gas station charges $0.20 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 $30? SOLUTION Because you are getting a car wash, you will pay $4.09 – $0.20 = $3.89 per gallon of gasoline. Let g be the amount (in gallons) of gasoline that you buy.

Example 5 STEP 1 Write a verbal model. Then write an inequality. g 3.89 + 10 < – 30

You can buy up to slightly more than 5 gallons of gasoline. Example 5 STEP 2 Solve the inequality. 3.89g + 10 ≤ 30 Write inequality. 3.89g ≤ 20 Subtract 10 from each side. g ≤ 5.14138. . . Divide each side by 3.89. ANSWER You can buy up to slightly more than 5 gallons of gasoline.

Example 5 CHECK You can use a table to check the reasonableness of your answer. The table shows that you will pay $29.45 for exactly 5 gallons of gasoline. Because $29.45 is less than $30, it is reasonable to conclude that you can buy slightly more than 5 gallons of gasoline.

Guided Practice WHAT IF? In Example 5, suppose that a car wash costs $11 and gasoline regularly costs $4.19 per gallon. What are the possible amounts (in gallons) of gasoline that you can buy? 7. Up to about 4.76 gallons ANSWER Camp Costs 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? at least $92.86/wk ANSWER

Lesson Quiz 1. Solve 3(x + 8) < 9. Graph your solution. ANSWER all real numbers less than –5

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

Lesson 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.