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# Integrated Math 2 Section 3-7

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Solving Inequalities

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• ### Integrated Math 2 Section 3-7

1. 1. Section 3-7 Solve Inequalities
2. 2. Essential Questions How do you solve and graph inequalities on a number line? How do you solve problems involving inequalities? Where you’ll see this: Communications, health, ﬁtness, hobbies, safety, business
3. 3. Vocabulary 1. Solve an Inequality: 2. Addition Property of Inequality:
4. 4. Vocabulary 1. Solve an Inequality: Find all values that make the inequality true 2. Addition Property of Inequality:
5. 5. Vocabulary 1. Solve an Inequality: Find all values that make the inequality true 2. Addition Property of Inequality:
6. 6. Vocabulary 1. Solve an Inequality: Find all values that make the inequality true 2. Addition Property of Inequality: If a < b, then a + c < b + c
7. 7. Vocabulary 1. Solve an Inequality: Find all values that make the inequality true 2. Addition Property of Inequality: If a < b, then a + c < b + c If a > b, then a + c > b + c
8. 8. Vocabulary 1. Solve an Inequality: Find all values that make the inequality true 2. Addition Property of Inequality: If a < b, then a + c < b + c If a > b, then a + c > b + c If you add to one side of an inequality, you add to the other
9. 9. Vocabulary 3. Multiplication and Division Properties of Inequality:
10. 10. Vocabulary 3. Multiplication and Division Properties of Inequality: If a < b and c > 0, then ac < bc
11. 11. Vocabulary 3. Multiplication and Division Properties of Inequality: If a < b and c > 0, then ac < bc If a > b and c > 0, then ac > bc
12. 12. Vocabulary 3. Multiplication and Division Properties of Inequality: If a < b and c > 0, then ac < bc If a > b and c > 0, then ac > bc If a < b and c < 0, then ac > bc
13. 13. Vocabulary 3. Multiplication and Division Properties of Inequality: If a < b and c > 0, then ac < bc If a > b and c > 0, then ac > bc If a < b and c < 0, then ac > bc If a > b and c < 0, then ac < bc
14. 14. Vocabulary 3. Multiplication and Division Properties of Inequality: If a < b and c > 0, then ac < bc If a > b and c > 0, then ac > bc If a < b and c < 0, then ac > bc If a > b and c < 0, then ac < bc This is the same for division!
15. 15. Vocabulary 3. Multiplication and Division Properties of Inequality: If a < b and c > 0, then ac < bc If a > b and c > 0, then ac > bc If a < b and c < 0, then ac > bc If a > b and c < 0, then ac < bc This is the same for division! This means that if we multiply or divide by a negative, we have to ﬂip the sign.
16. 16. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22
17. 17. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7
18. 18. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7 −n ≤ −2
19. 19. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7 −n ≤ −2 −1 −1
20. 20. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7 −n ≤ −2 −1 −1 n≥2
21. 21. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7 −n ≤ −2 −1 −1 n≥2 -3 -2 -1 0 1 2 3 4 5
22. 22. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7 −n ≤ −2 −1 −1 n≥2 -3 -2 -1 0 1 2 3 4 5
23. 23. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7 −n ≤ −2 −1 −1 n≥2 -3 -2 -1 0 1 2 3 4 5
24. 24. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7 +4 +4 −n ≤ −2 −1 −1 n≥2 -3 -2 -1 0 1 2 3 4 5
25. 25. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7 +4 +4 −n ≤ −2 13x > 26 −1 −1 n≥2 -3 -2 -1 0 1 2 3 4 5
26. 26. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7 +4 +4 −n ≤ −2 13x > 26 −1 −1 13 13 n≥2 -3 -2 -1 0 1 2 3 4 5
27. 27. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7 +4 +4 −n ≤ −2 13x > 26 −1 −1 13 13 n≥2 x >2 -3 -2 -1 0 1 2 3 4 5
28. 28. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7 +4 +4 −n ≤ −2 13x > 26 −1 −1 13 13 n≥2 x >2 -3 -2 -1 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 4 5
29. 29. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7 +4 +4 −n ≤ −2 13x > 26 −1 −1 13 13 n≥2 x >2 -3 -2 -1 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 4 5
30. 30. Example 1 Solve and graph each inequality. a. 7 − n ≤ 5 b. 13x − 4 > 22 −7 −7 +4 +4 −n ≤ −2 13x > 26 −1 −1 13 13 n≥2 x >2 -3 -2 -1 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 4 5
31. 31. Example 1 Solve and graph each inequality. c. − 4x ≥ 16 d. − 2x + 3 ≤ 8
32. 32. Example 1 Solve and graph each inequality. c. − 4x ≥ 16 d. − 2x + 3 ≤ 8 −4 −4
33. 33. Example 1 Solve and graph each inequality. c. − 4x ≥ 16 d. − 2x + 3 ≤ 8 −4 −4 x ≤ −4
34. 34. Example 1 Solve and graph each inequality. c. − 4x ≥ 16 d. − 2x + 3 ≤ 8 −4 −4 x ≤ −4 -7 -6 -5 -4 -3 -2 -1 0 1
35. 35. Example 1 Solve and graph each inequality. c. − 4x ≥ 16 d. − 2x + 3 ≤ 8 −4 −4 x ≤ −4 -7 -6 -5 -4 -3 -2 -1 0 1
36. 36. Example 1 Solve and graph each inequality. c. − 4x ≥ 16 d. − 2x + 3 ≤ 8 −4 −4 x ≤ −4 -7 -6 -5 -4 -3 -2 -1 0 1
37. 37. Example 1 Solve and graph each inequality. c. − 4x ≥ 16 d. − 2x + 3 ≤ 8 −4 −4 −3 −3 x ≤ −4 -7 -6 -5 -4 -3 -2 -1 0 1
38. 38. Example 1 Solve and graph each inequality. c. − 4x ≥ 16 d. − 2x + 3 ≤ 8 −4 −4 −3 −3 −2x ≤ 5 x ≤ −4 -7 -6 -5 -4 -3 -2 -1 0 1
39. 39. Example 1 Solve and graph each inequality. c. − 4x ≥ 16 d. − 2x + 3 ≤ 8 −4 −4 −3 −3 −2x ≤ 5 x ≤ −4 −2 −2 -7 -6 -5 -4 -3 -2 -1 0 1
40. 40. Example 1 Solve and graph each inequality. c. − 4x ≥ 16 d. − 2x + 3 ≤ 8 −4 −4 −3 −3 −2x ≤ 5 x ≤ −4 −2 −2 5 x≥− 2 -7 -6 -5 -4 -3 -2 -1 0 1
41. 41. Example 1 Solve and graph each inequality. c. − 4x ≥ 16 d. − 2x + 3 ≤ 8 −4 −4 −3 −3 −2x ≤ 5 x ≤ −4 −2 −2 5 x≥− 2 -7 -6 -5 -4 -3 -2 -1 0 1 -3 -2 -1 0 1
42. 42. Example 1 Solve and graph each inequality. c. − 4x ≥ 16 d. − 2x + 3 ≤ 8 −4 −4 −3 −3 −2x ≤ 5 x ≤ −4 −2 −2 5 x≥− 2 -7 -6 -5 -4 -3 -2 -1 0 1 -3 -2 -1 0 1
43. 43. Example 1 Solve and graph each inequality. c. − 4x ≥ 16 d. − 2x + 3 ≤ 8 −4 −4 −3 −3 −2x ≤ 5 x ≤ −4 −2 −2 5 x≥− 2 -7 -6 -5 -4 -3 -2 -1 0 1 -3 -2 -1 0 1
44. 44. When multiplying/dividing by a negative:
45. 45. When multiplying/dividing by a negative: You are multiplying by the opposite, so use the opposite sign.
46. 46. Example 2 Matt Mitarnowski’s Rentals charges \$39 per day, plus \$.42 per mile driven. If Fuzzy Jeff rents a car for one day, what possible distances can he drive and keep his total rental charge to a maximum of \$80?
47. 47. Example 2 Matt Mitarnowski’s Rentals charges \$39 per day, plus \$.42 per mile driven. If Fuzzy Jeff rents a car for one day, what possible distances can he drive and keep his total rental charge to a maximum of \$80? m = # miles
48. 48. Example 2 Matt Mitarnowski’s Rentals charges \$39 per day, plus \$.42 per mile driven. If Fuzzy Jeff rents a car for one day, what possible distances can he drive and keep his total rental charge to a maximum of \$80? m = # miles 39 + .42m
49. 49. Example 2 Matt Mitarnowski’s Rentals charges \$39 per day, plus \$.42 per mile driven. If Fuzzy Jeff rents a car for one day, what possible distances can he drive and keep his total rental charge to a maximum of \$80? m = # miles 39 + .42m ≤ 80
50. 50. Example 2 Matt Mitarnowski’s Rentals charges \$39 per day, plus \$.42 per mile driven. If Fuzzy Jeff rents a car for one day, what possible distances can he drive and keep his total rental charge to a maximum of \$80? m = # miles 39 + .42m ≤ 80 −39 −39
51. 51. Example 2 Matt Mitarnowski’s Rentals charges \$39 per day, plus \$.42 per mile driven. If Fuzzy Jeff rents a car for one day, what possible distances can he drive and keep his total rental charge to a maximum of \$80? m = # miles 39 + .42m ≤ 80 −39 −39 .42m ≤ 41
52. 52. Example 2 Matt Mitarnowski’s Rentals charges \$39 per day, plus \$.42 per mile driven. If Fuzzy Jeff rents a car for one day, what possible distances can he drive and keep his total rental charge to a maximum of \$80? m = # miles 39 + .42m ≤ 80 −39 −39 .42m ≤ 41 .42 .42
53. 53. Example 2 Matt Mitarnowski’s Rentals charges \$39 per day, plus \$.42 per mile driven. If Fuzzy Jeff rents a car for one day, what possible distances can he drive and keep his total rental charge to a maximum of \$80? m = # miles 39 + .42m ≤ 80 −39 −39 13 .42m ≤ 41 m ≤ 97 .42 .42 21
54. 54. Example 2 Matt Mitarnowski’s Rentals charges \$39 per day, plus \$.42 per mile driven. If Fuzzy Jeff rents a car for one day, what possible distances can he drive and keep his total rental charge to a maximum of \$80? m = # miles 39 + .42m ≤ 80 −39 −39 13 .42m ≤ 41 m ≤ 97 .42 .42 21 Jeff can drive up to 97 miles.
55. 55. Homework
56. 56. Homework p. 134 #1-11 all, 13-45 odd “Behold the turtle. He makes progress only when he sticks his neck out.” - James Bryant Conant