Solving Inequalities

1. Section 3-7
Solve Inequalities

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. Vocabulary
1. Solve an Inequality:
2. Addition Property of Inequality:

4. Vocabulary
1. Solve an Inequality: Find all values that make the
inequality true
2. Addition Property of Inequality:

5. Vocabulary
1. Solve an Inequality: Find all values that make the
inequality true
2. Addition Property of Inequality:

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. 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. 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. Vocabulary
3. Multiplication and Division Properties of Inequality:

10. Vocabulary
3. Multiplication and Division Properties of Inequality:
If a < b and c > 0, then ac < bc

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

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

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

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

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

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

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

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. When multiplying/dividing by a negative:

45. When multiplying/dividing by a negative:
You are multiplying by the opposite, so use the
opposite sign.

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. 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. 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. 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. 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. 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. 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. 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. 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. Homework

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