1. 3-4 Solving Two-Step and Multi-Step Inequalities Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

2. 3-4 Solving Two-Step and Multi-Step Inequalities Warm Up Solve each equation. 1. 2x – 5 = –17 2. Solve each inequality and graph the solutions. 4. 3. 5 < t + 9 –6 14 t > –4 a ≤ –8

3. 3-4 Solving Two-Step and Multi-Step Inequalities 4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x – 5) + 4(x – 2) = 12. 5.0 Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. California Standards

4. 3-4 Solving Two-Step and Multi-Step Inequalities Inequalities that contain more than one operation require more than one step to solve. Use inverse operations to undo the operations in the inequality one at a time.

5. 3-4 Solving Two-Step and Multi-Step Inequalities Additional Example 1A: Solving Multi-Step Inequalities Solve the inequality and graph the solutions. 45 + 2b > 61 –45 2b > 16 b > 8 0246810 12 14 16 18 20 Since 45 is added to 2b, subtract 45 from both sides to undo the addition. Since b is multiplied by 2, divide both sides by 2 to undo the multiplication. The solution set is { b:b > 8 }.

6. 3-4 Solving Two-Step and Multi-Step Inequalities 8 – 3y ≥ 29 –8 –3y ≥ 21 y ≤ –7 Since 8 is added to –3y, subtract 8 from both sides to undo the addition. Since y is multiplied by –3, divide both sides by –3 to undo the multiplication. Change ≥ to ≤. –10 –8 –6–4 –2 0246810 –7 Additional Example 1B: Solving Multi-Step Inequalities Solve the inequality and graph the solutions. The solution set is { y:y  –7 }.

7. 3-4 Solving Two-Step and Multi-Step Inequalities Subtracting a number is the same as adding its opposite. 7 – 2t = 7 + (–2t) Remember!

8. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 1a Solve the inequality and graph the solutions. Check your answer. –12 ≥ 3x + 6 – 6 –18 ≥ 3x –6 ≥ x Since 6 is added to 3x, subtract 6 from both sides to undo the addition. Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. –10 –8 –6–4 –2 0246810 The solution set is { x:x ≤ –6 }.

9. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 1a Continued Solve the inequality and graph the solutions. Check your answer. Check –12 ≥ 3x + 6 Check the endpoint, – 6 – 12 = 3(x) + 6 – 12 – 18 + 6 – 12 3( – 6) + 6 – 12 Check a number less than – 6. – 12 ≥ 3(x) + 6 – 12 ≥ 3( – 7) + 6 – 12 ≥ – 21 + 6 – 12 ≥ – 15

10. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 1b Solve the inequality and graph the solutions. Check your answer. x < –11 –5 x + 5 < –6 –20 –12–8–4 –16 0 –11 Since x + 5 is divided by –2, multiply both sides by –2 to undo the division. Change > to <. Since 5 is added to x, subtract 5 from both sides to undo the addition. The solution set is { x:x < –11 }.

11. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 1b Continued Solve the inequality and graph the solutions. Check your answer. Check Check the endpoint, – 11 3 Check a number less than – 11. 4 > 3

12. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 1c Solve the inequality and graph the solutions. Check your answer. 1 – 2n ≥ 21 –1 –2n ≥ 20 n ≤ –10 Since 1 – 2n is divided by 3, multiply both sides by 3 to undo the division. Since 1 is added to −2n, subtract 1 from both sides to undo the addition. Since n is multiplied by −2, divide both sides by −2 to undo the multiplication. Change ≥ to ≤. The solution set is { n:n ≤ –10 }.

13. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 1c Continued Solve the inequality and graph the solutions. Check your answer. Check Check the endpoint, – 10 7 Check a number less than – 10. 9 ≥ 7 –10 –20 –12–8–4 –16 0

14. 3-4 Solving Two-Step and Multi-Step Inequalities To solve more complicated inequalities, you may first need to simplify the expressions on one or both sides.

15. 3-4 Solving Two-Step and Multi-Step Inequalities Additional Example 2A: Simplifying Before Solving Inequalities Solve the inequality and graph the solutions. 2 – (–10) > –4t 12 > –4t –3 < t (or t > –3) Combine like terms. Since t is multiplied by –4, divide both sides by –4 to undo the multiplication. Change > to <. –3 –10 –8 –6–4 –2 02468 10 The solution set is { t:t > –3 }.

16. 3-4 Solving Two-Step and Multi-Step Inequalities Additional Example 2B: Simplifying Before Solving Inequalities Solve the inequality and graph the solutions. –4(2 – x) ≤ 8 −4(2 – x) ≤ 8 −4(2) − 4(−x) ≤ 8 –8 + 4x ≤ 8 +8 4x ≤ 16 x ≤ 4 Distribute –4 on the left side. Since –8 is added to 4x, add 8 to both sides. Since x is multiplied by 4, divide both sides by 4 to undo the multiplication. –10 –8 –6–4 –2 02468 10 The solution set is { x:x ≤ 4 }.

17. 3-4 Solving Two-Step and Multi-Step Inequalities Additional Example 2C: Simplifying Before Solving Inequalities Solve the inequality and graph the solutions. 4f + 3 > 2 –3 4f > –1 Multiply both sides by 6, the LCD of the fractions. Distribute 6 on the left side. Since 3 is added to 4f, subtract 3 from both sides to undo the addition.

18. 3-4 Solving Two-Step and Multi-Step Inequalities 4f > –1 Since f is multiplied by 4, divide both sides by 4 to undo the multiplication. 0 Additional Example 2C Continued The solution set is { f: f > }. Solve the inequality and graph the solutions.

19. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 2a Solve the inequality and graph the solutions. Check your answer. – 5 > – 5 2m > 20 m > 10 Since 5 is added to 2m, subtract 5 from both sides to undo the addition. Simplify 5 2. Since m is multiplied by 2, divide both sides by 2 to undo the multiplication. 0246810 12 14 16 18 20 2m + 5 > 5 2 2m + 5 > 25 The solution set is { m:m > 10 }.

20. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 2a Continued Solve the inequality and graph the solutions. Check your answer. 2m + 5 > 5 2 Check Check the endpoint, 10 2m + 5 = 5 2 2(10) + 5 5 2 20 + 5 25 25 Check a number greater than 10. 2m + 5 > 5 2 2(11) + 5 > 5 2 22 + 5 > 25 27 > 25

21. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 2b Solve the inequality and graph the solutions. Check your answer. 3 + 2(x + 4) > 3 3 + 2x + 8 > 3 2x + 11 > 3 – 11 2x > –8 x > –4 Distribute 2 on the left side. Combine like terms. Since 11 is added to 2x, subtract 11 from both sides to undo the addition. Since x is multiplied by 2, divide both sides by 2 to undo the multiplication. –10 –8 –6–4 –2 02468 10 The solution set is { x:x > –4 }.

22. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 2b Continued Solve the inequality and graph the solutions. Check your answer. 3 + 2(x + 4) > 3 Check Check the endpoint, – 4 3 3 + 2(x + 4) = 3 3 + 2( – 4 + 4) 3 3 + 2(0) 3 Check a number greater than – 4. 7 > 3 3 + 2(x + 4) > 3 3 + 2( – 2 + 4) > 3 3 + 2(2) > 3

23. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 2c Solve the inequality and graph the solutions. Check your answer. 5 < 3x – 2 +2 + 2 7 < 3x Multiply both sides by 8, the LCD of the fractions. Distribute 8 on the right side. Since 2 is subtracted from 3x, add 2 to both sides to undo the subtraction.

24. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 2c Continued Solve the inequality and graph the solutions. Check your answer. 7 < 3x 468 2 10 0 Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. The solution set is { x:x > }.

25. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 2c Continued Solve the inequality and graph the solutions. Check Check the endpoint, Check a number greater than

26. 3-4 Solving Two-Step and Multi-Step Inequalities Additional Example 3: Application To rent a certain vehicle, Rent-A-Ride charges $55.00 per day with unlimited miles. The cost of renting a similar vehicle at We Got Wheels is $38.00 per day plus $0.20 per mile. For what number of miles is the cost at Rent-A-Ride less than the cost at We Got Wheels? Let m represent the number of miles. The cost for Rent-A-Ride should be less than that of We Got Wheels. Cost at Rent-A- Ride must be less than daily cost at We Got Wheels plus $0.20 times # of miles. 55 < 38 +0.20  m

27. 3-4 Solving Two-Step and Multi-Step Inequalities 85 < m Since 38 is added to 0.20m, subtract 38 from both sides to undo the addition. Since m is multiplied by 0.20, divide both sides by 0.20 to undo the multiplication. Rent-A-Ride costs less when the number of miles is more than 85. Additional Example 3 Continued 55 < 38 + 0.20m –38 55 < 38 + 0.20m 17 < 0.20m

28. 3-4 Solving Two-Step and Multi-Step Inequalities Check Additional Example 3 Continued Check the endpoint, 85. 55 38 + 17 55 55 = 38 + 0.20m 55 38 + 0.20(85) Check a number greater than 85. 55 < 38 + 18 55 < 56 55 < 38 + 0.20(90) 55 < 38 + 0.20m

29. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 3 The average of Jim ’ s two test scores must be at least 90 to make an A in the class. Jim got a 95 on his first test. What grades can Jim get on his second test to make an A in the class? Let x represent the test score needed. The average score is the sum of each score divided by 2. First test score 17 second test score divided by number of scores is greater than or equal to total score (95 + x) x)  2 ≥ 90

30. 3-4 Solving Two-Step and Multi-Step Inequalities Check It Out! Example 3 Continued The score on the second test must be 85 or higher. Since 95 is added to x, subtract 95 from both sides to undo the addition. 95 + x ≥ 180 –95 x ≥ 85 Since 95 + x is divided by 2, multiply both sides by 2 to undo the division.

31. 3-4 Solving Two-Step and Multi-Step Inequalities Check Check It Out! Example 3 Continued Check the end point, 85. Check a number greater than 85. 90.5 ≥ 90 90

32. 3-4 Solving Two-Step and Multi-Step Inequalities Lesson Quiz: Part I Solve each inequality and graph the solutions. 1. 13 – 2x ≥ 21x ≤ –4 2. –11 + 2 < 3p p > –3 3. 2 3 < – 2(3 – t) t > 7 4.

33. 3-4 Solving Two-Step and Multi-Step Inequalities Lesson Quiz: Part II 5. A video store has two movie rental plans. Plan A includes a $25 membership fee plus $1.25 for each movie rental. Plan B costs $40 for unlimited movie rentals. For what number of movie rentals is plan B less than plan A? more than 12 movies