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Section 3-7
Solve Inequalities
Essential Questions
How do you solve and graph inequalities on a
number line?
How do you solve problems involving
inequali...
Vocabulary
1. Solve an Inequality:


2. Addition Property of Inequality:
Vocabulary
1. Solve an Inequality: Find all values that make the
    inequality true
2. Addition Property of Inequality:
Vocabulary
1. Solve an Inequality: Find all values that make the
    inequality true
2. Addition Property of Inequality:
Vocabulary
1. Solve an Inequality: Find all values that make the
    inequality true
2. Addition Property of Inequality:
 ...
Vocabulary
1. Solve an Inequality: Find all values that make the
    inequality true
2. Addition Property of Inequality:
 ...
Vocabulary
1. Solve an Inequality: Find all values that make the
    inequality true
2. Addition Property of Inequality:
 ...
Vocabulary
3. Multiplication and Division Properties of Inequality:
Vocabulary
3. Multiplication and Division Properties of Inequality:
              If a < b and c > 0, then ac < bc
Vocabulary
3. Multiplication and Division Properties of Inequality:
              If a < b and c > 0, then ac < bc
       ...
Vocabulary
3. Multiplication and Division Properties of Inequality:
              If a < b and c > 0, then ac < bc
       ...
Vocabulary
3. Multiplication and Division Properties of Inequality:
              If   a < b and c > 0, then ac < bc
     ...
Vocabulary
3. Multiplication and Division Properties of Inequality:
              If a < b and c > 0, then ac < bc
       ...
Vocabulary
3. Multiplication and Division Properties of Inequality:
           If a < b and c > 0, then ac < bc
          ...
Example 1
      Solve and graph each inequality.

a. 7 − n ≤ 5                 b. 13x − 4 > 22
Example 1
      Solve and graph each inequality.

a. 7 − n ≤ 5                 b. 13x − 4 > 22
 −7      −7
Example 1
      Solve and graph each inequality.

a. 7 − n ≤ 5                 b. 13x − 4 > 22
 −7      −7
    −n ≤ −2
Example 1
      Solve and graph each inequality.

a. 7 − n ≤ 5                 b. 13x − 4 > 22
 −7      −7
    −n ≤ −2
   ...
Example 1
      Solve and graph each inequality.

a. 7 − n ≤ 5                 b. 13x − 4 > 22
 −7      −7
    −n ≤ −2
   ...
Example 1
        Solve and graph each inequality.

 a. 7 − n ≤ 5                  b. 13x − 4 > 22
  −7      −7
     −n ≤ ...
Example 1
        Solve and graph each inequality.

 a. 7 − n ≤ 5                  b. 13x − 4 > 22
  −7      −7
     −n ≤ ...
Example 1
        Solve and graph each inequality.

 a. 7 − n ≤ 5                  b. 13x − 4 > 22
  −7      −7
     −n ≤ ...
Example 1
        Solve and graph each inequality.

 a. 7 − n ≤ 5                  b. 13x − 4 > 22
  −7      −7           ...
Example 1
        Solve and graph each inequality.

 a. 7 − n ≤ 5                  b. 13x − 4 > 22
  −7      −7           ...
Example 1
        Solve and graph each inequality.

 a. 7 − n ≤ 5                  b. 13x − 4 > 22
  −7      −7           ...
Example 1
        Solve and graph each inequality.

 a. 7 − n ≤ 5                  b. 13x − 4 > 22
  −7      −7           ...
Example 1
        Solve and graph each inequality.

 a. 7 − n ≤ 5                  b. 13x − 4 > 22
  −7      −7           ...
Example 1
        Solve and graph each inequality.

 a. 7 − n ≤ 5                  b. 13x − 4 > 22
  −7      −7           ...
Example 1
        Solve and graph each inequality.

 a. 7 − n ≤ 5                  b. 13x − 4 > 22
  −7      −7           ...
Example 1
     Solve and graph each inequality.

c. − 4x ≥ 16                d. − 2x + 3 ≤ 8
Example 1
     Solve and graph each inequality.

c. − 4x ≥ 16                d. − 2x + 3 ≤ 8
    −4 −4
Example 1
     Solve and graph each inequality.

c. − 4x ≥ 16                d. − 2x + 3 ≤ 8
    −4 −4
     x ≤ −4
Example 1
          Solve and graph each inequality.

 c. − 4x ≥ 16                    d. − 2x + 3 ≤ 8
        −4 −4
     ...
Example 1
          Solve and graph each inequality.

 c. − 4x ≥ 16                    d. − 2x + 3 ≤ 8
        −4 −4
     ...
Example 1
          Solve and graph each inequality.

 c. − 4x ≥ 16                    d. − 2x + 3 ≤ 8
        −4 −4
     ...
Example 1
          Solve and graph each inequality.

 c. − 4x ≥ 16                    d. − 2x + 3 ≤ 8
        −4 −4      ...
Example 1
          Solve and graph each inequality.

 c. − 4x ≥ 16                    d. − 2x + 3 ≤ 8
        −4 −4      ...
Example 1
          Solve and graph each inequality.

 c. − 4x ≥ 16                    d. − 2x + 3 ≤ 8
        −4 −4      ...
Example 1
          Solve and graph each inequality.

 c. − 4x ≥ 16                    d. − 2x + 3 ≤ 8
        −4 −4      ...
Example 1
          Solve and graph each inequality.

 c. − 4x ≥ 16                    d. − 2x + 3 ≤ 8
        −4 −4      ...
Example 1
          Solve and graph each inequality.

 c. − 4x ≥ 16                    d. − 2x + 3 ≤ 8
        −4 −4      ...
Example 1
          Solve and graph each inequality.

 c. − 4x ≥ 16                    d. − 2x + 3 ≤ 8
        −4 −4      ...
When multiplying/dividing by a negative:
When multiplying/dividing by a negative:



You are multiplying by the opposite, so use the
                opposite sign.
Example 2
Matt Mitarnowski’s Rentals charges $39 per day, plus
$.42 per mile driven. If Fuzzy Jeff rents a car for one
day...
Example 2
Matt Mitarnowski’s Rentals charges $39 per day, plus
$.42 per mile driven. If Fuzzy Jeff rents a car for one
day...
Example 2
Matt Mitarnowski’s Rentals charges $39 per day, plus
$.42 per mile driven. If Fuzzy Jeff rents a car for one
day...
Example 2
Matt Mitarnowski’s Rentals charges $39 per day, plus
$.42 per mile driven. If Fuzzy Jeff rents a car for one
day...
Example 2
Matt Mitarnowski’s Rentals charges $39 per day, plus
$.42 per mile driven. If Fuzzy Jeff rents a car for one
day...
Example 2
Matt Mitarnowski’s Rentals charges $39 per day, plus
$.42 per mile driven. If Fuzzy Jeff rents a car for one
day...
Example 2
Matt Mitarnowski’s Rentals charges $39 per day, plus
$.42 per mile driven. If Fuzzy Jeff rents a car for one
day...
Example 2
Matt Mitarnowski’s Rentals charges $39 per day, plus
$.42 per mile driven. If Fuzzy Jeff rents a car for one
day...
Example 2
Matt Mitarnowski’s Rentals charges $39 per day, plus
$.42 per mile driven. If Fuzzy Jeff rents a car for one
day...
Homework
Homework



           p. 134 #1-11 all, 13-45 odd




“Behold the turtle. He makes progress only when he
    sticks his n...
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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

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