1. Solving Multistep Inequalities
10-4
Pre-Algebra

2. Warm Up Solve. 1. 6x + 36 = 2x 2. 4x – 13 = 15 + 5x7 8 3 16 11 16
x = –

3. Learn to solve two-step inequalities and graph the solutions of an inequality on a number line.

4. Solving a multistep inequality uses the same inverse operations as solving a multistep equation. Multiplying or dividing the inequality by a negative number reverses the inequality symbol.

5. Example Solve and graph. A. 4x + 1 > 13 4x + 1 > 13
– 1 – 1 Subtract 1 from both sides.
4x > 12 4x 4
> 12
Divide both sides by 4.
x > 3

6. Example B. –7 < 3x + 8 – 8 – 8 Subtract 8 from both sides.
– 15 3
< 3x
Divide both sides by 3.
–5 < x

7. Example
C. -9x + 7  25
–9x + 7  25
– 7 – 7 Subtract 7 from both sides.
–9x  18
–9x –9  18
Divide each side by –9; change  to .
x  –2

8. Try This Solve and graph. A. 5x + 2 > 12 5x + 2 > 12
– 2 – 2 Subtract 2 from both sides.
5x > 10 5x 5
> 10
Divide both sides by 5.
x > 2

9. Try This B. –5 < 2x + 9 – 9 – 9 Subtract 9 from both sides.
– 14 2
< 2x
Divide both sides by 2.
–7 < x

10. Try This
C. -4x + 2  18
–4x + 2  18
– 2 – 2 Subtract 2 from both sides.
–4x  16
–4x –4  16
Divide each side by –4; change  to .
x  –4

11. Example Solve and graph. A. 10x + 21 – 4x < –15
6x < – Combine like terms.
– – 21 Subtract 21 from both sides.
6x < –36 6x 6
<
–36
Divide both sides by 6.
x < –6

12. Example B. +  2x 5 3 4 9 10 +  2x 5 3 4 9 10 20( + )  20( ) 2x 5 3
+  2x 5 3 4 9 10
20( + )  20( ) 2x 5 3 4 9 10
Multiply by LCD, 20.
20( ) + 20( )  20( ) 2x 5 3 4 9 10
8x + 15  18
– 15 – Subtract 15 from both sides.
8x  3

13. Example Continued 8x  3  8x 8 3 Divide both sides by 8. x  3 8 3 8 3 8

14. Example C. 8x + 8 > 11x – 1 8x + 8 > 11x – 1
– 8x – 8x Subtract 8x from both sides.
8 > 3x – 1
Add 1 to each side.
9 > 3x 9 3
> 3x
Divide both sides by 3.
3 > x

15. Try This Solve and graph. A. 15x + 30 – 5x < –10
10x < – Combine like terms.
– – 30 Subtract 30 from both sides.
10x < –40
10x 10
<
–40
Divide both sides by 10.
x < –4

16. Try This B. +  3x 5 1 4 10 +  3x 5 1 4 10 20( + )  20( ) 3x 5 1 4
+  3x 5 1 4 10
20( + )  20( ) 3x 5 1 4 10
Multiply by LCD, 20.
20( ) + 20( )  20 ( ) 3x 5 1 4 10
12x + 5  10
– 5 – Subtract 5 from both sides.
12x  5

17. Try This Continued 12x  5  12x 12 5 Divide both sides by 12. x  55 12

18. Try This C. 4x + 3 > 8x – 1 4x + 3 > 8x – 1
– 4x – 4x Subtract 4x from both sides.
3 > 4x – 1
Add 1 to each side.
4 > 4x 4
> 4x
Divide both sides by 4.
1 > x

19. Example: Business Application
A school’s Spanish club is selling bumper stickers. They bought 100 bumper stickers for $55, and they have to give the company 15 cents for every sticker sold. If they plan to sell each bumper sticker for $1.25, how many do they have to sell to make a profit?
Let R represent the revenue and C represent the cost. In order for the Spanish club to make a profit, the revenue must be greater than the cost.
R > C

20. Example Continued
The revenue from selling x bumper stickers at $1.25 each is 1.25x. The cost of selling x bumper stickers is the fixed cost plus the unit cost times the number of bumper stickers sold, or x. Substitute the expressions for R and C.
1.25x > x
Let x represent the number of bumper stickers sold. Fixed cost is $55. Unit cost is 15 cents.

21. Example Continued 1.25x > 55 + 0.15x
Subtract 0.15x from both sides.
– 0.15x – 0.15x
1.10x > 55
1.10x
1.10 55
>
Divide both sides by 1.10.
x > 50
The Spanish club must sell more than 50 bumper stickers to make a profit.

22. Try This
A school’s Spanish club is selling bumper stickers. They bought 200 bumper stickers for $45, and they have to give the company 25 cents for every sticker sold. If they plan to sell each bumper sticker for $2.50, how many do they have to sell to make a profit?
Let R represent the revenue and C represent the cost. In order for the Spanish club to make a profit, the revenue must be greater than the cost.
R > C

23. Try This Continued
The revenue from selling x bumper stickers at $2.50 each is 2.5x. The cost of selling x bumper stickers is the fixed cost plus the unit cost times the number of bumper stickers sold, or x. Substitute the expressions for R and C.
2.5x > x
Let x represent the number of bumper stickers sold. Fixed cost is $45. Unit cost is 25 cents.

24. Try This Continued 2.5x > 45 + 0.25x
Subtract 0.25x from both sides.
– 0.25x – 0.25x
2.25x > 45
2.25x
2.25 45
>
Divide both sides by 2.25.
x > 20
The Spanish club must sell more than 20 bumper stickers to make a profit.

25. Lesson Quiz: Part 1 Solve and graph. 1. 4x – 6 > 10
2. 7x + 9 < 3x – 15
3. w – 3w < 32
4. w + 
x > 4
x < –6
w > –16 2 3 1 4 1 2
w  3 8 3 8

26. Lesson Quiz: Part 2
5. Antonio has budgeted an average of $45 a month for entertainment. For the first five months of the year he has spent $48, $39, $60, $48, and $33. How much 1 can Antonio spend in the sixth month without exceeding his average budget?
no more than $42