Solving Inequalities by Multiplication and Division

1. SECTION 5-2
Solving Inequalities by Multiplication and Division
Monday, September 23, 13

2. Essential Question
How do we solve and graph inequalities with multiplication or
division?
Monday, September 23, 13

3. Question
How does the process of solving an inequality compare to the
process of solving an equation?
Monday, September 23, 13

4. FLIPPING THE INEQUALITY
Monday, September 23, 13

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. 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. 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. 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. 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. 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. Example 2
Solve the inequality and graph the solution.
a. −
3
5
x ≥ 6
Monday, September 23, 13

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. 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. 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. 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. 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. 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. Example 2
Solve the inequality and graph the solution.
b.12k ≥ 60
Monday, September 23, 13

19. Example 2
Solve the inequality and graph the solution.
b.12k ≥ 60
12k
12
≥
60
12
Monday, September 23, 13

20. Example 2
Solve the inequality and graph the solution.
b.12k ≥ 60
12k
12
≥
60
12
k ≤ 5
Monday, September 23, 13

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. 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. 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. 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. Example 2
Solve the inequality and graph the solution.
c. − 8q <136
Monday, September 23, 13

26. Example 2
Solve the inequality and graph the solution.
c. − 8q <136
−8q
−8
<
136
−8
Monday, September 23, 13

27. Example 2
Solve the inequality and graph the solution.
c. − 8q <136
−8q
−8
<
136
−8
q > −17
Monday, September 23, 13

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. 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. 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. 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. Example 2
Solve the inequality and graph the solution.
d. −
2
3
m ≤ −10
Monday, September 23, 13

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Problem Sets
Monday, September 23, 13

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. 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