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# Algebra 1B Section 5-2

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Solving Inequalities by Multiplication and Division

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### Algebra 1B Section 5-2

1. 1. SECTION 5-2 Solving Inequalities by Multiplication and Division Monday, September 23, 13
2. 2. Essential Question How do we solve and graph inequalities with multiplication or division? Monday, September 23, 13
3. 3. Question How does the process of solving an inequality compare to the process of solving an equation? Monday, September 23, 13
4. 4. FLIPPING THE INEQUALITY Monday, September 23, 13
5. 5. FLIPPING THE INEQUALITY Whenever you multiply or divide each side of an equation by a negative number, you must ﬂip the inequality sign! Monday, September 23, 13
6. 6. Example 1 Matt Mitarnowski walks at a rate of mile an hour. He knows that he is at least 6 miles from Lake Shecky. How long will it take him to get there? Write and solve an inequality to ﬁnd the time. 3 4 t = time to get there Monday, September 23, 13
7. 7. Example 1 Matt Mitarnowski walks at a rate of mile an hour. He knows that he is at least 6 miles from Lake Shecky. How long will it take him to get there? Write and solve an inequality to ﬁnd the time. 3 4 t = time to get there 3 4 t ≥ 6 Monday, September 23, 13
8. 8. Example 1 Matt Mitarnowski walks at a rate of mile an hour. He knows that he is at least 6 miles from Lake Shecky. How long will it take him to get there? Write and solve an inequality to ﬁnd the time. 3 4 t = time to get there 3 4 t ≥ 6 4 3 • 3 4 t ≥ 6• 4 3 Monday, September 23, 13
9. 9. Example 1 Matt Mitarnowski walks at a rate of mile an hour. He knows that he is at least 6 miles from Lake Shecky. How long will it take him to get there? Write and solve an inequality to ﬁnd the time. 3 4 t = time to get there 3 4 t ≥ 6 4 3 • 3 4 t ≥ 6• 4 3 t ≥ 8 Monday, September 23, 13
10. 10. Example 1 Matt Mitarnowski walks at a rate of mile an hour. He knows that he is at least 6 miles from Lake Shecky. How long will it take him to get there? Write and solve an inequality to ﬁnd the time. 3 4 t = time to get there 3 4 t ≥ 6 4 3 • 3 4 t ≥ 6• 4 3 t ≥ 8 It will take Matt at least 8 hours to walk to Lake Shecky. Monday, September 23, 13
11. 11. Example 2 Solve the inequality and graph the solution. a. − 3 5 x ≥ 6 Monday, September 23, 13
12. 12. Example 2 Solve the inequality and graph the solution. a. − 3 5 x ≥ 6 − 5 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 3 5 x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≥ 6( ) − 5 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Monday, September 23, 13
13. 13. Example 2 Solve the inequality and graph the solution. a. − 3 5 x ≥ 6 − 5 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 3 5 x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≥ 6( ) − 5 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ x ≤ −10 Monday, September 23, 13
14. 14. Example 2 Solve the inequality and graph the solution. a. − 3 5 x ≥ 6 − 5 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 3 5 x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≥ 6( ) − 5 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ x ≤ −10 x | x ≤ −10{ } Monday, September 23, 13
15. 15. Example 2 Solve the inequality and graph the solution. a. − 3 5 x ≥ 6 − 5 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 3 5 x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≥ 6( ) − 5 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ x ≤ −10 x | x ≤ −10{ } -10 -9 -8 -7 -6-11-12-13-14 Monday, September 23, 13
16. 16. Example 2 Solve the inequality and graph the solution. a. − 3 5 x ≥ 6 − 5 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 3 5 x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≥ 6( ) − 5 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ x ≤ −10 x | x ≤ −10{ } -10 -9 -8 -7 -6-11-12-13-14 Monday, September 23, 13
17. 17. Example 2 Solve the inequality and graph the solution. a. − 3 5 x ≥ 6 − 5 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 3 5 x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≥ 6( ) − 5 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ x ≤ −10 x | x ≤ −10{ } -10 -9 -8 -7 -6-11-12-13-14 Monday, September 23, 13
18. 18. Example 2 Solve the inequality and graph the solution. b.12k ≥ 60 Monday, September 23, 13
19. 19. Example 2 Solve the inequality and graph the solution. b.12k ≥ 60 12k 12 ≥ 60 12 Monday, September 23, 13
20. 20. Example 2 Solve the inequality and graph the solution. b.12k ≥ 60 12k 12 ≥ 60 12 k ≤ 5 Monday, September 23, 13
21. 21. Example 2 Solve the inequality and graph the solution. b.12k ≥ 60 12k 12 ≥ 60 12 k ≤ 5 k | k ≤ 5{ } Monday, September 23, 13
22. 22. Example 2 Solve the inequality and graph the solution. b.12k ≥ 60 12k 12 ≥ 60 12 k ≤ 5 k | k ≤ 5{ } 5 6 7 8 94321 Monday, September 23, 13
23. 23. Example 2 Solve the inequality and graph the solution. b.12k ≥ 60 12k 12 ≥ 60 12 k ≤ 5 k | k ≤ 5{ } 5 6 7 8 94321 Monday, September 23, 13
24. 24. Example 2 Solve the inequality and graph the solution. b.12k ≥ 60 12k 12 ≥ 60 12 k ≤ 5 k | k ≤ 5{ } 5 6 7 8 94321 Monday, September 23, 13
25. 25. Example 2 Solve the inequality and graph the solution. c. − 8q <136 Monday, September 23, 13
26. 26. Example 2 Solve the inequality and graph the solution. c. − 8q <136 −8q −8 < 136 −8 Monday, September 23, 13
27. 27. Example 2 Solve the inequality and graph the solution. c. − 8q <136 −8q −8 < 136 −8 q > −17 Monday, September 23, 13
28. 28. Example 2 Solve the inequality and graph the solution. c. − 8q <136 −8q −8 < 136 −8 q > −17 q | q > −17{ } Monday, September 23, 13
29. 29. Example 2 Solve the inequality and graph the solution. c. − 8q <136 −8q −8 < 136 −8 q > −17 q | q > −17{ } -17 -16 -15 -14 -13-18-19-20-21 Monday, September 23, 13
30. 30. Example 2 Solve the inequality and graph the solution. c. − 8q <136 −8q −8 < 136 −8 q > −17 q | q > −17{ } -17 -16 -15 -14 -13-18-19-20-21 Monday, September 23, 13
31. 31. Example 2 Solve the inequality and graph the solution. c. − 8q <136 −8q −8 < 136 −8 q > −17 q | q > −17{ } -17 -16 -15 -14 -13-18-19-20-21 Monday, September 23, 13
32. 32. Example 2 Solve the inequality and graph the solution. d. − 2 3 m ≤ −10 Monday, September 23, 13
33. 33. Example 2 Solve the inequality and graph the solution. d. − 2 3 m ≤ −10 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 2 3 m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ −10( ) − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Monday, September 23, 13
34. 34. Example 2 Solve the inequality and graph the solution. d. − 2 3 m ≤ −10 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 2 3 m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ −10( ) − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ m ≥15 Monday, September 23, 13
35. 35. Example 2 Solve the inequality and graph the solution. d. − 2 3 m ≤ −10 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 2 3 m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ −10( ) − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ m ≥15 m | m ≥15{ } Monday, September 23, 13
36. 36. Example 2 Solve the inequality and graph the solution. d. − 2 3 m ≤ −10 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 2 3 m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ −10( ) − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ m ≥15 m | m ≥15{ } 15 16 17 18 1914131211 Monday, September 23, 13
37. 37. Example 2 Solve the inequality and graph the solution. d. − 2 3 m ≤ −10 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 2 3 m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ −10( ) − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ m ≥15 m | m ≥15{ } 15 16 17 18 1914131211 Monday, September 23, 13
38. 38. Example 2 Solve the inequality and graph the solution. d. − 2 3 m ≤ −10 − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 2 3 m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ −10( ) − 3 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ m ≥15 m | m ≥15{ } 15 16 17 18 1914131211 Monday, September 23, 13
39. 39. Example 3 Maggie Brann and friends order a pizza. Maggie eats 2 of the 16 slices and pays \$2.80 for her share.Assuming that Maggie has paid at least her fair share, what is the most the pizza cost? Identify a variable, then set up and solve an inequality. Interpret your solution. Monday, September 23, 13
40. 40. Example 3 Maggie Brann and friends order a pizza. Maggie eats 2 of the 16 slices and pays \$2.80 for her share.Assuming that Maggie has paid at least her fair share, what is the most the pizza cost? Identify a variable, then set up and solve an inequality. Interpret your solution. c = cost of pizza Monday, September 23, 13
41. 41. Example 3 Maggie Brann and friends order a pizza. Maggie eats 2 of the 16 slices and pays \$2.80 for her share.Assuming that Maggie has paid at least her fair share, what is the most the pizza cost? Identify a variable, then set up and solve an inequality. Interpret your solution. c = cost of pizza 2 16 c ≤ 2.8 Monday, September 23, 13
42. 42. Example 3 Maggie Brann and friends order a pizza. Maggie eats 2 of the 16 slices and pays \$2.80 for her share.Assuming that Maggie has paid at least her fair share, what is the most the pizza cost? Identify a variable, then set up and solve an inequality. Interpret your solution. c = cost of pizza 2 16 c ≤ 2.8 1 8 c ≤ 2.8 Monday, September 23, 13
43. 43. Example 3 Maggie Brann and friends order a pizza. Maggie eats 2 of the 16 slices and pays \$2.80 for her share.Assuming that Maggie has paid at least her fair share, what is the most the pizza cost? Identify a variable, then set up and solve an inequality. Interpret your solution. c = cost of pizza 2 16 c ≤ 2.8 1 8 c ≤ 2.8 8 1 8 c ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ 2.8( )8 Monday, September 23, 13
44. 44. Example 3 Maggie Brann and friends order a pizza. Maggie eats 2 of the 16 slices and pays \$2.80 for her share.Assuming that Maggie has paid at least her fair share, what is the most the pizza cost? Identify a variable, then set up and solve an inequality. Interpret your solution. c = cost of pizza 2 16 c ≤ 2.8 1 8 c ≤ 2.8 8 1 8 c ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ 2.8( )8 c ≤ 22.4 Monday, September 23, 13
45. 45. Example 3 Maggie Brann and friends order a pizza. Maggie eats 2 of the 16 slices and pays \$2.80 for her share.Assuming that Maggie has paid at least her fair share, what is the most the pizza cost? Identify a variable, then set up and solve an inequality. Interpret your solution. c = cost of pizza 2 16 c ≤ 2.8 1 8 c ≤ 2.8 8 1 8 c ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ 2.8( )8 c ≤ 22.4 c | c ≤ 22.4{ } Monday, September 23, 13
46. 46. Example 3 Maggie Brann and friends order a pizza. Maggie eats 2 of the 16 slices and pays \$2.80 for her share.Assuming that Maggie has paid at least her fair share, what is the most the pizza cost? Identify a variable, then set up and solve an inequality. Interpret your solution. c = cost of pizza 2 16 c ≤ 2.8 1 8 c ≤ 2.8 8 1 8 c ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≤ 2.8( )8 c ≤ 22.4 The pizza cost at most \$22.40 c | c ≤ 22.4{ } Monday, September 23, 13
47. 47. Summarizer Using complete sentences, explain when you would switch the sign in an inequality.What is the reason as to why this occurs? Monday, September 23, 13
48. 48. Problem Sets Monday, September 23, 13
49. 49. Problem Sets Problem Set 1: p. 293 #1-9 Problem Set 2: p. 293 #12-27 multiplies of 3, 31-33 all, 63-68 all Monday, September 23, 13
50. 50. Problem Sets Problem Set 1: p. 293 #1-9 Problem Set 2: p. 293 #12-27 multiplies of 3, 31-33 all, 63-68 all "If you can ﬁnd a path with no obstacles, it probably doesn't lead anywhere." - Frank A. Clark Monday, September 23, 13