1. Solving Multi-Step Inequalities
Section 3-4

2. Goals Goal Rubric To solve multi-step inequalities.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
None

4. Solving Multi-Step Inequalities
When you solved multi-step equations, you used the order of operations in reverse to isolate the variable.
You can use the same process when solving multi-step inequalities.
Use inverse operations in the inverse order to undo the operations in the inequality one at a time.

5. Example: Solve the inequality and graph the solutions. 45 + 2b > 61
Since 45 is added to 2b, subtract 45 from both sides to undo the addition.
45 + 2b > 61
– –45
2b > 16
Since b is multiplied by 2, divide both sides by 2 to undo the multiplication.
b > 8 2 4 6 8 10 12 14 16 18 20

6. Very Important
If both sides of an inequality are multiplied or divided by a negative number, the inequality symbol must be reversed.
Remember!

7. Example: Solve the inequality and graph the solutions. 8 – 3y ≥ 29
Since 8 is added to –3y, subtract 8 from both sides to undo the addition.
8 – 3y ≥ 29
– –8
–3y ≥ 21
Since y is multiplied by –3, divide both sides by –3 to undo the multiplication. Change ≥ to ≤.
y ≤ –7
–10 –8 –6 –4 –2 2 4 6 8 10 –7

8. Subtracting a number is the same as adding its opposite.
7 – 2t = 7 + (–2t)
Remember!

9. Your Turn: Solve the inequality and graph the solutions. –12 ≥ 3x + 6
Since 6 is added to 3x, subtract 6 from both sides to undo the addition.
–12 ≥ 3x + 6
– – 6
–18 ≥ 3x
Since x is multiplied by 3, divide both sides by 3 to undo the multiplication.
–6 ≥ x
–10 –8 –6 –4 –2 2 4 6 8 10

10. Your Turn: Solve the inequality and graph the solutions.
Since x is divided by –2, multiply both sides by –2 to undo the division. Change > to <.
–5 –5
x + 5 < –6
Since 5 is added to x, subtract 5 from both sides to undo the addition.
x < –11
–20
–12 –8 –4
–16
–11

11. Your Turn: Solve the inequality and graph the solutions.
Since 1 – 2n is divided by 3, multiply both sides by 3 to undo the division.
1 – 2n ≥ 21
Since 1 is added to −2n, subtract 1 from both sides to undo the addition.
– –1
–2n ≥ 20
Since n is multiplied by −2, divide both sides by −2 to undo the multiplication. Change ≥ to ≤.
n ≤ –10
–10
–20
–12 –8 –4
–16

12. Solving Multi-Step Inequalities
To solve more complicated inequalities, you may first need to simplify the expressions on one or both sides by using the order of operations, combining like terms, or using the Distributive Property.

13. Example: Solve the inequality and graph the solutions.
Combine like terms.
Since t is multiplied by –4, divide both sides by –4 to undo the multiplication. Change > to <.
–3 < t
(or t > –3) –3
–10 –8 –6 –4 –2 2 4 6 8 10

14. Example: Solve the inequality and graph the solutions. –4(2 – x) ≤ 8
Distribute –4 on the left side.
−4(2) − 4(−x) ≤ 8
Since –8 is added to 4x, add 8 to both sides.
–8 + 4x ≤ 8
4x ≤ 16
Since x is multiplied by 4, divide both sides by 4 to undo the multiplication.
x ≤ 4
–10 –8 –6 –4 –2 2 4 6 8 10

15. Example: Solve the inequality and graph the solutions. 4f + 3 > 2
Multiply both sides by 6, the LCD of the fractions.
Distribute 6 on the left side.
4f + 3 > 2
Since 3 is added to 4f, subtract 3 from both sides to undo the addition.
–3 –3
4f > –1

16. Example: Continued 4f > –1
Since f is multiplied by 4, divide both sides by 4 to undo the multiplication.

17. Your Turn: Solve the inequality and graph the solutions.
2m + 5 > 52
Simplify 52.
2m + 5 > 25
– 5 > – 5
Since 5 is added to 2m, subtract 5 from both sides to undo the addition.
2m > 20
m > 10
Since m is multiplied by 2, divide both sides by 2 to undo the multiplication. 2 4 6 8 10 12 14 16 18 20

18. Your Turn: Solve the inequality and graph the solutions.
3 + 2(x + 4) > 3
Distribute 2 on the left side.
3 + 2(x + 4) > 3
3 + 2x + 8 > 3
Combine like terms.
2x + 11 > 3
Since 11 is added to 2x, subtract 11 from both sides to undo the addition.
– 11 – 11
2x > –8
Since x is multiplied by 2, divide both sides by 2 to undo the multiplication.
x > –4
–10 –8 –6 –4 –2 2 4 6 8 10

19. Your Turn: Solve the inequality and graph the solutions. 5 < 3x – 2
Multiply both sides by 8, the LCD of the fractions.
Distribute 8 on the right side.
5 < 3x – 2
Since 2 is subtracted from 3x, add 2 to both sides to undo the subtraction.
7 < 3x

20. Your Turn: Continued Solve the inequality and graph the solutions.
7 < 3x
Since x is multiplied by 3, divide both sides by 3 to undo the multiplication. 4 6 8 2 10

21. daily cost at We Got Wheels
Example: Application
To rent a certain vehicle, Rent-A-Ride charges $55.00 per day with unlimited miles. The cost of renting a similar vehicle at We Got Wheels is $38.00 per day plus $0.20 per mile. For what number of miles in the cost at Rent-A-Ride less than the cost at We Got Wheels?
Let m represent the number of miles. The cost for Rent-A-Ride should be less than that of We Got Wheels.
Cost at Rent-A-Ride
must be less than
daily cost at We Got Wheels
plus
$0.20
per mile
times
# of miles. 55
< 38 +
0.20  m

22. Example: Continued 55 < 38 + 0.20m
Since 38 is added to 0.20m, subtract 8 from both sides to undo the addition.
–38 –38
55 < m
17 < 0.20m
Since m is multiplied by 0.20, divide both sides by 0.20 to undo the multiplication.
85 < m
Rent-A-Ride costs less when the number of miles is more than 85.

23. is greater than or equal to
Your Turn:
The average of Jim’s two test scores must be at least 90 to make an A in the class. Jim got a 95 on his first test. What grades can Jim get on his second test to make an A in the class?
Let x represent the test score needed. The average score is the sum of each score divided by 2.
First test score
plus
second test score
divided by
number of scores
is greater than or equal to
total score
(95 + x)  2 ≥ 90

24. Your Turn: Continued 95 + x ≥ 180 –95 –95 x ≥ 85
Since 95 + x is divided by 2, multiply both sides by 2 to undo the division.
95 + x ≥ 180
Since 95 is added to x, subtract 95 from both sides to undo the addition.
– –95
x ≥ 85
The score on the second test must be 85 or higher.

25. Solving Inequalities with Variables on Both Sides
Some inequalities have variable terms on both sides of the inequality symbol. You can solve these inequalities like you solved equations with variables on both sides.
Use the properties of inequality to “collect” all the variable terms on one side and all the constant terms on the other side.

26. Example: Solve the inequality and graph the solutions. y ≤ 4y + 18
To collect the variable terms on one side, subtract y from both sides.
Since 18 is added to 3y, subtract 18 from both sides to undo the addition.
– – 18
–18 ≤ 3y
Since y is multiplied by 3, divide both sides by 3 to undo the multiplication.
–6 ≤ y (or y  –6)
–10 –8 –6 –4 –2 2 4 6 8 10

27. Example: Solve the inequality and graph the solutions.
4m – 3 < 2m + 6
To collect the variable terms on one side, subtract 2m from both sides.
–2m – 2m
2m – 3 <
Since 3 is subtracted from 2m, add 3 to both sides to undo the subtraction
2m <
Since m is multiplied by 2, divide both sides by 2 to undo the multiplication. 4 5 6

28. Your Turn: Solve the inequality and graph the solutions. 4x ≥ 7x + 6
To collect the variable terms on one side, subtract 7x from both sides.
–3x ≥
x ≤ –2
Since x is multiplied by –3, divide both sides by –3 to undo the multiplication. Change ≥ to ≤.
–10 –8 –6 –4 –2 2 4 6 8 10

29. Your Turn: Solve the inequality and graph the solutions.
5t + 1 < –2t – 6
5t + 1 < –2t – 6
+2t t
7t + 1 < –6
To collect the variable terms on one side, add 2t to both sides.
Since 1 is added to 7t, subtract 1 from both sides to undo the addition.
– 1 < –1
7t < –7
Since t is multiplied by 7, divide both sides by 7 to undo the multiplication.
7t < –7
t < –1 –5 –4 –3 –2 –1 1 2 3 4 5

30. Example: Application
The Home Cleaning Company charges $312 to power-wash the siding of a house plus $12 for each window. Power Clean charges $36 per window, and the price includes power-washing the siding. How many windows must a house have to make the total cost from The Home Cleaning Company less expensive than Power Clean?
Let w be the number of windows.

31. Example: Continued 312 + 12 • w < 36 • w 312 + 12w < 36w
Home
Cleaning
Company
siding charge
plus
$12 per window
# of windows
is less
than
Power
Clean
cost per window
# of
windows.
times
• w < • w
w < 36w
– 12w –12w
To collect the variable terms, subtract 12w from both sides.
312 < 24w
Since w is multiplied by 24, divide both sides by 24 to undo the multiplication.
13 < w
The Home Cleaning Company is less expensive for houses with more than 13 windows.

32. Your Turn:
A-Plus Advertising charges a fee of $24 plus $0.10 per flyer to print and deliver flyers. Print and More charges $0.25 per flyer. For how many flyers is the cost at A-Plus Advertising less than the cost of Print and More?
Let f represent the number of flyers printed.
plus
$0.10
per flyer
is less
than
# of flyers.
A-Plus
Advertising
fee of $24
Print and
More’s cost
# of flyers
times
• f < • f

33. Your Turn: Continued 24 + 0.10f < 0.25f –0.10f –0.10f 24 < 0.15f
To collect the variable terms, subtract 0.10f from both sides.
< 0.15f
Since f is multiplied by 0.15, divide both sides by 0.15 to undo the multiplication.
160 < f (or f > 160)
More than 160 flyers must be delivered to make A-Plus Advertising the lower cost company.

34. Simplifying Both Sides to Solve Inequalities
You may need to simplify one or both sides of an inequality before solving it. Look for like terms to combine and places to use the Distributive Property.

35. Example: Solve the inequality and graph the solutions.
2(k – 3) > 6 + 3k – 3
Distribute 2 on the left side of the inequality.
2(k – 3) > 3 + 3k
2k + 2(–3) > 3 + 3k
2k – 6 > 3 + 3k
To collect the variable terms, subtract 2k from both sides.
–2k – 2k
–6 > 3 + k
Since 3 is added to k, subtract 3 from both sides to undo the addition.
–3 –3
–9 > k
or k < - 9
–12 –9 –6 –3 3

36. Example: Solve the inequality and graph the solution.
0.9y ≥ 0.4y – 0.5
0.9y ≥ 0.4y – 0.5
To collect the variable terms, subtract 0.4y from both sides.
–0.4y –0.4y
0.5y ≥ – 0.5
0.5y ≥ –0.5
Since y is multiplied by 0.5, divide both sides by 0.5 to undo the multiplication.
y ≥ –1 –5 –4 –3 –2 –1 1 2 3 4 5

37. Your Turn: Solve the inequality and graph the solutions.
5(2 – r) ≥ 3(r – 2)
Distribute 5 on the left side of the inequality and distribute 3 on the right side of the inequality.
5(2 – r) ≥ 3(r – 2)
5(2) – 5(r) ≥ 3(r) + 3(–2)
10 – 5r ≥ 3r – 6
Since 6 is subtracted from 3r, add 6 to both sides to undo the subtraction.
16 − 5r ≥ 3r
Since 5r is subtracted from 16 add 5r to both sides to undo the subtraction.
+ 5r +5r
≥ 8r
Since r is multiplied by 8, divide both sides by 8 to undo the multiplication.
2 ≥ r (or r ≤ 2) –6 –2 2 –4 4

38. Your Turn: Solve the inequality and graph the solutions.
0.5x – x < 0.3x + 6
2.4x – 0.3 < 0.3x + 6
Simplify.
2.4x – 0.3 < 0.3x + 6
Since 0.3 is subtracted from 2.4x, add 0.3 to both sides.
2.4x < 0.3x + 6.3
Since 0.3x is added to 6.3, subtract 0.3x from both sides.
–0.3x –0.3x
2.1x <
Since x is multiplied by 2.1, divide both sides by 2.1.
x < 3 –5 –4 –3 –2 –1 1 2 3 4 5

39. Joke Time Why did the Easter egg hide? He was a little chicken!
What do you get if you cross rabbits and termites?
Bugs bunnies!
Why does a lobster never share?
Because it’s shellfish!

40. Assignment
3-4 Exercises Pg. 203 – 205: #8 – 54 even