1. 13.1
Writing Inequalities
How can you use inequalities to represent real-world constraints or conditions?

2. Warm Up Solve. 1. –21z + 12 = –27z 2. –12n – 18 = –6n 3. 12y – 56 = 8y
4. –36k + 9 = –18k
z = –2
n = –3
y = 14 1 2
k =

3. Symbol Meaning Word Phrases
An inequality states that two quantities
either are not equal or may not be
equal. An inequality uses one of the
following symbols:
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

4. Additional Example 1: Writing Inequalities
Write an inequality for each situation.
A. There are at least 15 people in the waiting room.
“At least” means greater
than or equal to.
number of people ≥ 15
B. The tram attendant will allow no more
than 60 people on the tram.
“No more than” means
less than or equal to.
number of people ≤ 60

5. Check It Out: Example 1
Write an inequality for each situation.
A. There are at most 10 gallons of gas in the tank.
“At most” means less
than or equal to.
gallons of gas ≤ 10
B. There is at least 10 yards of fabric left.
“At least” means
greater than or equal to.
yards of fabric ≥ 10

6. An inequality that contains a variable is an algebraic inequality
An inequality that contains a variable is an algebraic inequality. A value of the variable that makes the inequality true is a solution of the inequality.
An inequality may have more than one solution. Together, all of the solutions are called the solution set.
You can graph the solutions of an inequality on a number line. If the variable is “greater than” or “less than” a number, then that number is indicated with an open circle.

7. a > 5 b ≤ 3 This open circle shows that 5 is not a solution.
If the variable is “greater than or equal to” or “less than or equal to” a number, that number is indicated with a closed circle.
This closed circle shows that 3 is a solution.
b ≤ 3

8. Additional Example 2: Graphing Simple Inequalities
Graph each inequality.
A. n < 3
3 is not a solution, so
draw an open circle at 3. Shade the line to the left of 3.
–3 –2 –
B. a ≥ –4
–4 is a solution, so
draw a closed circle
at –4. Shade the line
to the right of –4.
–6 –4 –

9. draw a closed circle at 2. Shade the line to the left of 2.
Check It Out: Example 2
Graph each inequality.
A. p ≤ 2
2 is a solution, so
draw a closed circle at 2. Shade the line to the left of 2.
–3 –2 –
B. e > –2
–2 is not a solution, so
draw an open circle
at –2. Shade the line
to the right of –2.
–3 –2 –

10. x is either greater than 3 or less than–1.
A compound inequality is the result of combining two inequalities. The words and and or are used to describe how the two parts are related.
x > 3 or x < –1
–2 < y and y < 4
x is either greater than 3 or less than–1.
y is both greater than –2 and less than y is between –2 and 4.
The compound inequality –2 < y and y < 4 can be written as –2 < y < 4.
Writing Math

11. Additional Example 3A: Graphing Compound Inequalities
Graph each compound inequality.
m ≤ –2 or m > 1
First graph each inequality separately.
m ≤ –2
m > 1 • 2 4 6 – 2 4 6 –2 –4 –6
Then combine the graphs. 1 2 3 4 5 6 –
The solutions of m ≤ –2 or m > 1 are the combined solutions of m ≤ –2 or m > 1.

12. Additional Example 3B: Graphing Compound Inequalities
Graph each compound inequality
–3 < b ≤ 0
–3 < b ≤ 0 can be written as the inequalities
–3 < b and b ≤ 0. Graph each inequality separately.
–3 < b
b ≤ 0 • 2 4 6 –2 –4 –6 2 4 6 –
Then combine the graphs. Remember that –3 < b ≤ 0 means that b is between –3 and 0, and includes 0. 1 2 3 4 5 6 –

13. -3 < b is the same as b > -3.
Reading Math

14. Graph each compound inequality.
Check It Out: Example 3A
Graph each compound inequality.
w < 2 or w ≥ 4
First graph each inequality separately.
w < 2
W ≥ 4 2 4 6 – 2 4 6 –2 –4 –6
Then combine the graphs. 1 2 3 4 5 6 –
The solutions of w < 2 or w ≥ 4 are the combined solutions of w < 2 or w ≥ 4.

15. • º Check It Out: Example 3B Graph each compound inequality
5 > g ≥ –3 can be written as the inequalities
5 > g and g ≥ –3. Graph each inequality separately.
5 > g
g ≥ –3 • 2 4 6 –2 –4 –6 2 4 6 –
Then combine the graphs. Remember that > g ≥ –3 means that g is between 5 and –3, and includes –3. 1 2 3 4 5 6 –

16. ADDITIONAL EXAMPLE 1 A b ≥ –4 Sample check: –1 ≥ –4 B –3 > s
Graph the solutions of each inequality. Check the solutions. A
b ≥ –4
Sample check: –1 ≥ –4 B
–3 > s
Sample check: –3 > –5

17. ADDITIONAL EXAMPLE 2 A
Write an inequality that represents the phrase “y minus 3 is less than or equal to –5.” Then graph the inequality.
y – 3 ≤ –5 B
The temperature of the river is greater than –1 °C. Write and graph an inequality to represent this situation.
t > –1

18. Write an inequality for each situation.
1. No more than 220 people are in the theater.
2. There are at least a dozen eggs left.
3. Fewer than 14 people attended the meeting.
people in the theater ≤ 220
number of eggs ≥ 12
people attending the meeting < 14

19. 13.1 LESSON QUIZ Graph each inequality. 1. a ≤ –2 2. n < 4 3.4.
t ≤ 3

20. Write an inequality that matches the number line model
Write an inequality that matches the number line model. Use x for the variable. 5.
x ≤ –5 6.
x > –1 7.
x ≥ –0.5 8.
x < 2

21. How can you use inequalities to represent real-world constraints or conditions?
Sample answer:
You can choose a letter to represent the variable value in the situation and then use one of the inequality symbols to describe its range of values.