Objective The student will be able to:

1. ObjectiveThe student will be able to: solve inequalities. MFCR Lesson 1-7 9-25-14

2. 9-25-14 Bellwork

3. Section 1.7—Linear Inequalities in One Variable • Copy Key concepts from p. 94 • Read examples 1 & 2

4. 1.7—Linear Inequalities in One Variable • A linear inequality in one variable can be written in the form ax + b < 0, ax + b> 0, ax + b ≤ 0, or ax + b ≥ 0

5. Properties of Inequalities • If a < b, then a + c < b + c. • If a < b, then a – c < b – c. • If c is positive and a < b, then ac < bc and a/c < b/c. • If c is negative and a < b, then ac > bc and a/c > b/c. *Properties 3 and 4 indicate that if we multiply or divide an inequality by a negative value, the direction of the inequality sign must be reversed.

6. o 3 4 5 1) Solve 5m - 8 > 12 + 8 + 8 5m > 20 5 5 m > 4 5(4) – 8 = 12 • Draw “the river” • Add 8 to both sides • Simplify • Divide both sides by 5 • Simplify • Check your answer • Graph the solution

7. o -3 -2 -1 2) Solve 12 - 3a > 18 - 12 - 12 -3a > 6 -3 -3 a < -2 12 - 3(-2) = 18 • Draw “the river” • Subtract 12 from both sides • Simplify • Divide both sides by -3 • Simplify (Switch the inequality!) • Check your answer • Graph the solution

8. Answer Now Which graph shows the solution to 2x - 10 ≥ 4? • .

9. o 4 5 6 3) Solve 5m - 4 < 2m + 11 -2m -2m 3m - 4 < 11 + 4 + 4 3m < 15 3 3 m < 5 5(5) – 4 = 2(5) + 11 • Draw “the river” • Subtract 2m from both sides • Simplify • Add 4 to both sides • Simplify • Divide both sides by 3 • Simplify • Check your answer • Graph the solution

10. ● -8 -7 -6 4) Solve 2r - 18 ≤ 5r + 3 -2r -2r -18 ≤ 3r + 3 - 3 - 3 -21 ≤ 3r 3 3 -7 ≤ r or r ≥ -7 2(-7) – 18 = 5(-7) + 3 • Draw “the river” • Subtract 2r from both sides • Simplify • Subtract 3 from both sides • Simplify • Divide both sides by 3 • Simplify • Check your answer • Graph the solution

11. Answer Now 6) Solve -2x + 6 ≥ 3x - 4 • x ≥ -2 • x ≤ -2 • x ≥ 2 • x ≤ 2

12. o 6 7 8 -14p -14p 12p – 20 > 64 + 20 + 20 12p > 84 12 12 p > 7 26(7) – 20 = 14(7) + 64 • Draw “the river” • Subtract 14p from both sides • Simplify • Add 20 to both sides • Simplify • Divide both sides by 12 • Simplify • Check your answer • Graph the solution 5) Solve 26p - 20 > 14p + 64

13. Answer Now What are the values of x if 3(x + 4) - 5(x - 1) < 5? • x < -6 • x > -6 • x < 6 • x > 6

14. ObjectivesThe student will be able to: 1. solve compound inequalities. 2. graph the solution sets of compound inequalities.

15. Compound Inequalities To solve a compound inequality, isolate the variable x in the “middle.” The operations performed on the middle portion of the inequality must also be performed on the left-hand side and right-hand side. Ex. 1: -2 ≤ 3x + 1 < 5 Ex. 2: -8 < 5x – 3 ≤ 12

16. What is the difference between and and or? A B AND means intersection -what do the two items have in common? OR means union -if it is in one item, it is in the solution A B

17. o o ● ● ● o 2 2 2 2 3 3 3 3 4 4 4 4 1) Graph x < 4 and x ≥ 2 a) Graph x < 4 b) Graph x ≥ 2 c) Combine the graphs d) Where do they intersect?

18. o o ● ● 2 2 2 2 3 3 3 3 4 4 4 4 2) Graph x < 2 or x ≥ 4 a) Graph x < 2 b) Graph x ≥ 4 c) Combine the graphs

19. o o -3 -2 -1 Answer Now 3) Which inequalities describe the following graph? • y > -3 or y < -1 • y > -3 and y < -1 • y ≤ -3 or y ≥ -1 • y ≥ -3 and y ≤ -1

20. o o 6 7 8 4) Graph the compound inequality 6 < m < 8 When written this way, it is the same thing as 6 < m AND m < 8 It can be rewritten as m > 6 and m < 8 and graphed as previously shown, however, it is easier to graph everything between 6 and 8!

21. Answer Now 5) Which is equivalent to-3 < y < 5? • y > -3 or y < 5 • y > -3 and y < 5 • y < -3 or y > 5 • y < -3 and y > 5

22. Answer Now 6) Which is equivalent to x > -5 and x ≤ 1? • -5 < x ≤ 1 • -5 > x ≥ 1 • -5 > x ≤ 1 • -5 < x ≥ 1

23. o o o o -6 -6 1 1 -3 -3 4 4 0 0 7 7 7) 2x < -6 and 3x ≥ 12 ● Solve each inequality for x Graph each inequality Combine the graphs Where do they intersect? They do not! x cannot be greater than or equal to 4 and less than -3 No Solution!! ●

24. 8) Graph 3 < 2m – 1 < 9 Remember, when written like this, it is an AND problem! 3 < 2m – 1 AND 2m – 1 < 9 Solve each inequality. Graph the intersection of 2 < m and m < 5. - 5 0 5

25. - 5 0 5 9) Graph x < 2 or x ≥ 4

26. 10) Graph x ≥ -1 or x ≤ 3 The whole line is shaded!!

27. Practice Problems

28. Solving a Compound Inequality Application • Beth received grades of 87%, 82%, 96%, and 79% on her last four algebra tests. To graduate with honors, she needs at least a B in the course. • What grade does she need to make on the 5th test to get a B in the course? Assume that the tests are weighted equally and that to earn a B the average of the test grades must be at least 80% but less than 90%. • Is it possible for Beth to earn an A in the course if an A requires an average of 90% or more?

29. Solving a Linear Inequality Application The number of registered passenger cars, N (in millions), in the U.S. has risen since 1960 according to the equation N = 2.5t + 64.4, where t represents the number of years after 1960 (t = 0 corresponds to 1960, t = 1 corresponds to 1961, etc.) • For what years was the number of registered passenger cars less than 89.4 million? • For what years was the number of registered passenger cars between 94.4 million and 101.9 million?

30. Exit Ticket – Hand in before leaving class. a. What are the solutions to -2 < x + 2 ≤ 5 b. Graph the solutions to the above compound inequality