1. AF4.1 Solve two-step linear inequalities in one variable over the rational numbers, interpret the solution or solutions in the context from which they arose.
California
Standards

2. Notes – explain this to your elbow partner
When you solved two-step equations, you used the order of operations in reverse to isolate the variable. You can use the same process when solving two-step inequalities.
Add or subtract first
Multiply or Divide last

3. Additional Example 1A: Solving Two-Step Inequalities
Notes
Solve and graph.
4x + 1 > 13
4x + 1 > 13
– 1 – 1 Since 1 is added to 4x, subtract 1 from both sides.
4x > 12 4x 4
> 12
Since x is multiplied by 4, divide both sides by 4.
x > 3

4. Is the coefficient negative
Is the coefficient negative? If yes the direction of the inequality symbol will change - Notes

5. Additional Example 1B: Solving Two-Step Inequalities
Notes
Solve and graph.
Is the coefficient negative?
–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

6. – 2 – 2 Subtract 2 from both sides.
Check It Out! Example 1A
Rally Coach A
Solve and graph.
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

7. – 2 – 2 Subtract 2 from both sides.
Check It Out! Example 1B
Rally Coach B
–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

8. Additional Example 2: Solving Inequalities That Contain Fractions5 3 4 9 10
Solve  and graph the solution.
+  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
Distributive Property.
8x + 15  18
– 15 – Since 15 is added to 8x, subtract 15 from both sides.
8x  3

9. Additional Example 2 Continued 8x 8 3
Since x multiplied by 8, divide both sides by 8.
x  3 8 3 8

10. – 5 – 5 Since 5 is added to 12x, subtract 5 from both sides.
Check It Out! Example 2
Notes
Solve  3x 5 1 4 10
Multiply by LCD, 20.
20( ) + 20( )  ( ) 3x 5 1 4 10
12x + 5  10
– 5 – Since 5 is added to 12x, subtract 5 from both sides.
12x  5

11. Check It Out! Example 2 Continued
Elbow Partners
12x  5 
12x 12 5
Since x is multiplied by 12, divide both sides by 12.
x  5 12 5 12
Is the coefficient negative

12. Additional Example 3: School 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?
In order for the Spanish club to make a profit, the revenue must be greater than the cost.
1.25x > x

13. Additional Example 3 Continued
1.25x > x
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.