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

2. Vocabulary
inequality
algebraic inequality
solution set

3. Symbol Meaning Word Phrases
An inequality compares two expressions using <, >, , or .
Symbol
Meaning
Word Phrases
<
> ≤ ≥
is less than
Fewer than, below
is greater than
More than, above
is less than or equal to
At most, no more than
is greater than or equal to
At least, no less than
An inequality that contains a variable is an algebraic inequality.

4. A solution of an inequality is any value of the variable that makes the inequality true. All of the solutions of an inequality are called the solution set.
You can graph the solution set on a number line. The symbols < and > indicate an open circle.

5. a > 5 b ≤ 3 This open circle shows that 5 is not a solution.
The symbols ≤ and ≥ indicate a closed circle.
This closed circle shows that 3 is a solution.
b ≤ 3

6. Additional Example 3: Graphing Inequalities
Graph each inequality.
A. –1 > y
Draw an open circle at –1. The solutions are all values of y less than –1, so shade the line to the left of –1.
–3 –2 – 12
B. z ≥ –2
Draw a closed circle
at –2 and all values of z greater than So shade to the right of –2 .
1 2
–3 –2 –

7. Check It Out! Example 3 Graph each inequality. A. n < 3
Draw an open circle at 3. The solutions are all values of n less than 3, so shade the line to the left of 3.
–3 –2 –
B. a ≥ –4
Draw a closed circle
at –4. The solutions are all values greater than –4, so shade to the right of –4.
–6 –4 –

8. 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.

9. When you multiply (or divide) both sides of an inequality by a negative number, you must reverse the inequality symbol to make the statement true.

10. Example 1: Solving Two-Step Inequalities
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

11.  Example 1 Continued Check
According to the graph 4 should be a solution and 2 should not be a solution.
x > 3
x > 3
Substitute 4 for x.
Substitute 2 for x. ? ?
4 > 3
2 > 3 
So 4 is a solution.
So 2 is not a solution.

12. Example 2: Solving Two-Step Inequalities
Solve and graph.
–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

13. – 2 – 2 Subtract 2 from both sides.
Check It Out! Example 3
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

14. Check It Out! Example 3 Continued
According to the graph 4 should be a solution and 1 should not be a solution.
x > 2
x > 2
Substitute 4 for x.
Substitute 1 for x. ? ?
4 > 2
1 > 2 
So 4 is a solution.
So 1 is not a solution.

15. – 2 – 2 Subtract 2 from both sides.
Check It Out! Example 4
–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

16. Example 5 4y + 10 < 18 < 10 + 4y 18 4y < 8 4 8 4y < < y
Solving and Graphing a Two-Step Inequality 4y + 10
< 18
Original inequality
< 10 + 4y 18
Subtract 10 from each side. 4y
< 8
Simplify. 4 8 4y
<
Divide each side by 4.
< y 2
Simplify. 1 3 4 5 16

17. Example 6
Multiple Choice Practice
You are organizing a bowling night for charity. Each ticket costs $10 and includes shoe rental. Door
prizes cost you $50. Which inequality describes the possible numbers x of people who need to attend for you to make a profit of at least $200? 15 x ≤ ≥ 25
SOLUTION
To find your profit, subtract the total costs from the total ticket sales. This amount should equal or exceed the minimum desired profit of $200. 17

18. Example 6 50 10x – 200 ≥ 250 ≥ 10x 10 10 25 ≥ x ANSWER
Multiple Choice Practice 50
10x –
200 ≥
Write an inequality.
Add 50 to each side.
250 ≥
10x
Divide each side by 10.
10 10 25 ≥ x
ANSWER
The correct answer is D. 18

19. Example 7 3x – 14 < – x 3x – 14 < – x – 3x – 3x < 14 – 4x
Combining Like Terms 3x – 14
< – x
Original inequality 3x – 14
< – x
Subtract 3x from each side. – 3x – 3x
Combine like terms.
< 14 – 4x
Divide each side by and
reverse the inequality symbol.
> 4 – 4x 14
Simplify. x 2 7
> 1 3 4 5 19

20. Guided Practice
Solve the inequality. Then graph the solution. 1. 15 7z + – ≥ 57
ANSWER z 6 – ≤ 2. 40
11n + 3n
<
ANSWER n 5 –
<

21. Solve the inequality. Then graph the solution.
Guided Practice
for Examples 1, 2, and 3
Solve the inequality. Then graph the solution. 3. 18 9y – 16
>
ANSWER
> y 9 2

22. Guided Practice 4. WHAT IF?
for Examples 1, 2, and 3
4. WHAT IF?
In Example 3, how many people need to attend for you to make a profit of at least $250?
ANSWER x ≥
30 people

23. Example 8: 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

24. Example 8 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.

25. Check It Out! Example 9
A school’s French 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?
In order for the French club to make a profit, the revenue must be greater than the cost.
2.5x > x

26. Check It Out! Example 9 Continued
2.5x > x
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 French club must sell more than 20 bumper stickers to make a profit.

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