1. Solve Multi-Step Inequalities
Warm Up
Lesson Presentation
Lesson Quiz

2. Warm-Up
1. Is 4 a solution of the inequality 2a – 1.5  6?
ANSWER
yes
2. Solve the equation of = 7. f 4
ANSWER –4
3. All hats sell for the same price at a swap meet. You buy one pair of sunglasses for $7 and 2 hats. How much did each hat cost?
ANSWER
$9.75

3. 3x – 7 < 8. Graph your solution. Solve
Example 1
3x – 7 < 8. Graph your solution.
Solve
3x – 7 < 8
Write original inequality.
3x < 15
Add 7 to each side.
x < 5
Divide each side by 3.
ANSWER
The solutions are all real numbers less than 5. Check by substituting a number less than 5 in the original inequality.

4. Example 1
CHECK
3x –7 < 8
Write original inequality. ?
3(0) – 7 < 8
Substitute 0 for x.
–7 < 8
Solution checks.

5. Example 2 Solve –0.6(x – 5)  15. –0.6(x – 5)  15 –0.6x + 3  15
Write original inequality.
–0.6x + 3  15
Distributive property
–0.6x  12
Subtract 3 from each side.
x 
–20
Divide each side by Reverse
inequality symbol. –

6. Guided Practice
Solve the inequality. Graph your solution.
2x – 1.
< – x
< –
ANSWER
–6y –16. 2.
< – y
> –
ANSWER 3. –1 4
( p –12) > –2
p < 20
ANSWER

7. 6x – 7 > 2x + 17. Graph your solution. Solve
Example 3
6x – 7 > 2x Graph your solution.
Solve
6x – 7 > 2x + 17
Write original inequality.
6x > 2x + 24
Add 7 to each side.
4x > 24
Subtract 2x from each side.
x > 6
Divide each side by 4.
ANSWER
The solutions are all real numbers greater than 6.

8. Solve the inequality, if possible. a. 14x + 5 < 7(2x – 3)
Example 4
Solve the inequality, if possible.
a x + 5 < 7(2x – 3)
SOLUTION
14x + 5 < 7(2x – 3) a.
Write original inequality.
14x + 5 < 14x – 21
Distributive property
5 < –21
Subtract 14x from each side.
ANSWER
There are no solutions because 5 < –21 is false.

9. All real numbers are solutions because –1 > –6 is true.
Example 4
b x – 1 > 6(2x – 1)
12x – 1 > 6(2x – 1)
Write original inequality.
12x – 1 > 12x – 6
Distributive property
–1 > –6
Subtract 12x from each side.
ANSWER
All real numbers are solutions because –1 > –6 is true.

10. Guided Practice
Solve the inequality, if possible. Graph your solution. 4.
5x – 12  3x – 4
x  4
ANSWER
(m + 5) < 5m + 17
ANSWER
no solution
1 – 8s –4(2s – 1) 6.
< –
all real numbers
ANSWER

11. Example 5
Car Wash
Use the sign shown. A gas station charges $0.20 less per gallon of gasoline if a customer also gets a car wash. What are the possible amounts (in gallons) of gasoline that you can buy if you also get a car wash and can spend at most $30?
SOLUTION
Because you are getting a car wash, you will pay $4.09 – $0.20 = $3.89 per gallon of gasoline. Let g be the amount (in gallons) of gasoline that you buy.

12. Example 5
STEP 1
Write a verbal model. Then write an inequality. g
3.89 + 10
< – 30

13. You can buy up to slightly more than 5 gallons of gasoline.
Example 5
STEP 2
Solve the inequality.
3.89g + 10 ≤ 30
Write inequality.
3.89g ≤ 20
Subtract 10 from each side.
g ≤
Divide each side by 3.89.
ANSWER
You can buy up to slightly more than 5 gallons of gasoline.

14. Example 5
CHECK
You can use a table to check the reasonableness of your answer. The table shows that you will pay $29.45 for exactly 5 gallons of gasoline. Because $29.45 is less than $30, it is reasonable to conclude that you can buy slightly more than 5 gallons of gasoline.

15. Guided Practice
WHAT IF? In Example 5, suppose that a car wash costs $11 and gasoline regularly costs $4.19 per gallon. What are the possible amounts (in gallons) of gasoline that you can buy? 7.
Up to about 4.76 gallons
ANSWER
Camp Costs
8. You are saving money for a summer camp that costs $1800. You have saved $500 so far, and you have 14 more weeks to save the total amount. What are the possible average amounts of money that you can save per week in order to have a total of at least $1800 saved?
at least $92.86/wk
ANSWER

16. Lesson Quiz
1. Solve 3(x + 8) < 9. Graph your solution.
ANSWER
all real numbers less than –5

17. Lesson Quiz
Solve the inequality, if possible.
k + 1 > 3(2k – 3)
ANSWER
all real numbers because 1 > –9 is true
(a – 2)< 5a – 14
ANSWER
no solutions because –10 < –14 is false

18. Lesson Quiz
4. A box of cat treats contains at least 50 treats. So far you have fed your cats 18 treats. If you want the box of treats to last 8 days, what are the possible average numbers of treats you can feed the cats per day?
ANSWER
an average of no more than 4 treats per day.