1. Graphing and Writing Inequalities
Holt Algebra 1
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 1

2. Warm Up Compare. Write <, >, or =. 1. –3 2 3. < 2. 6.5 6.3
1. –3 2 3.
<
>
> =
Tell whether the inequality x < 5 is true or false for the following values of x.
5. x = –10 T
6. x = 5 F
7. x = 4.99 T
8. x = T

3. Objectives Identify solutions of inequalities with one variable.
Write and graph inequalities with one variable.

4. Vocabulary Inequality Solution of an inequality Greater Than Less Than
Greater Than or Equal To
Less Than or Equal To
Compound Inequality
Intersection
Union

5. An inequality is a statement that two quantities are not equal
An inequality is a statement that two quantities are not equal. The quantities are compared by using the following signs: ≤
A ≤ B
A is less
than or
equal to B.
<
A < B
than B.
>
A > B
A is greater ≥
A ≥ B ≠
A ≠ B
A is not
A solution of an inequality is any value of the variable that makes the inequality true.

6. Example 1: Identifying Solutions of Inequalities
Describe the solutions of x – 6 ≥ 4 in words.
Solution?
–9 4 ≥ ? –3 –9 No
–6 4
4 4
6 4 x
x – 6
x – 6 ≥ 4
9.9 10
10.1 12 –6
3.9 4
4.1 6
Yes
When the value of x is a number less than 10, the value of x – 6 is less than 4.
When the value of x is 10, the value of x – 6 is equal to 4.
When the value of x is a number greater than 10, the value of x – 6 is greater than 4.
It appears that the solutions of x – 6 ≥ 4 are all real numbers greater than or equal to 10.

7. Describe the solutions of 2p > 8 in words.
Check It Out! Example 1
Describe the solutions of 2p > 8 in words. p –3
3.9 4
4.1 5 2p –6
7.8 8
8.2 10
2p > 8 ? ?
> ? ?
8 8 ? ? ?
–6 8
>
>
>
0 8
>
10 8
>
Solution? No No No No
Yes
Yes
When the value of p is a number less than 4, the value of 2p is less than 8.
When the value of p is 4, the value of 2p is equal to 8
When the value of p is a number greater than 4, the value of 2p is greater than 8.
It appears that the solutions of 2p > 8 are all real numbers greater than 4.

8. An inequality like 3 + x < 9 has too many solutions to list
An inequality like 3 + x < 9 has too many solutions to list. You can use a graph on a number line to show all the solutions.
The solutions are shaded and an arrow shows that the solutions continue past those shown on the graph. To show that an endpoint is a solution, draw a solid circle at the number. To show an endpoint is not a solution, draw an empty circle.

10. Example 2: Graphing Inequalities
Graph each inequality.
Draw a solid circle at .
A. m ≥
Shade all the numbers greater than and draw an arrow pointing to the right. 1 – 2 3
B. t < 5(–1 + 3)
Simplify.
t < 5(–1 + 3)
t < 5(2)
t < 10
Draw an empty circle at 10.
Shade all the numbers less than 10 and draw an arrow pointing to the left. –4 –2 2 4 6 8 10 12 –6 –8

11. Check It Out! Example 2 Graph each inequality. a. c > 2.5
Draw an empty circle at 2.5.
Shade in all the numbers greater than 2.5 and draw an arrow pointing to the right.
a. c > 2.5 –4 –3 –2 –1 1 2 3 4 5 6
2.5
b. 22 – 4 ≥ w
Draw a solid circle at 0.
22 – 4 ≥ w
Shade in all numbers less than 0 and draw an arrow pointing to the left.
4 – 4 ≥ w
0 ≥ w –4 –3 –2 –1 1 2 3 4 5 6
c. m ≤ –3
Draw a solid circle at –3. –4 –2 2 4 6 8 10 12 –6 –8 –3
Shade in all numbers less than –3 and draw an arrow pointing to the left.

12. Example 3: Writing an Inequality from a Graph
Write the inequality shown by each graph.
x < 2
Use any variable. The arrow points to the left, so use either < or ≤. The empty circle at 2 means that 2 is not a solution, so use <.
x ≥ –0.5
Use any variable. The arrow points to the right, so use either > or ≥. The solid circle at –0.5 means that –0.5 is a solution, so use ≥.

13. Check It Out! Example 3
Write the inequality shown by the graph.
Use any variable. The arrow points to the left, so use either < or ≤. The empty circle at 2.5 means that 2.5 is not a solution, so use so use <.
x < 2.5

14. Reading Math
“No more than” means “less than or equal to.”
“At least” means “greater than or equal to”.

15. Turn on the AC when temperature is at least 85°F
WARM UP
Ray’s dad told him not to turn on the air conditioner unless the temperature is at least 85°F. Define a variable and write an inequality for the temperatures at which Ray can turn on the air conditioner. Graph the solution.
Let t represent the temperatures at which Ray can turn on the air conditioner.
Turn on the AC when temperature
is at least
85°F t ≥ 85
Draw a solid circle at 85. Shade all numbers greater than 85 and draw an arrow pointing to the right.
t  85 75 80 85 90 70

16. Let w represent an employee’s wages.
WARM UP
A store’s employees earn at least $8.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Graph the solution.
Let w represent an employee’s wages.
What an employee earns
at least
$8.50 w ≥
8.50
w ≥ 8.5 4 6 8 10 12 −2 2 14 16 18
8.5

17. WARM UP
A cell phone plan offers free minutes for no more than 250 minutes per month. Define a variable and write an inequality for the possible number of free minutes. Graph the solution.
Let m = number of minutes
0 ≤ m ≤ 250
250

18. Warm Up
Sami has a gift card. She has already used $14 of the total value, which was $30. Write, solve, and graph an inequality to show how much more she can spend.

19. Example 2 Continued
Write an inequality.
Let g represent the remaining amount of money Sami can spend.
Amount remaining
plus
$30.
is at most
amount used g + 14 ≤ 30
g + 14 ≤ 30

20. The amount spent cannot be negative.
Example 2 Continued
g + 14 ≤ 30
Since 14 is added to g, subtract 14 from both sides to undo the addition.
– 14 – 14
g + 0 ≤ 16
g ≤ 16
Draw a solid circle at 0 and16. 2 4 6 8 10 12 14 16 18
Shade all numbers greater than 0 and less than 16.
The amount spent cannot be negative.

21. Let p represent the number of tubes of paint that Jill can buy.
Example 3: Application
Jill has a $20 gift card to an art supply store where 4 oz tubes of paint are $4.30 each after tax. What are the possible numbers of tubes that Jill can buy?
Let p represent the number of tubes of paint that Jill can buy.
$4.30
times
number of tubes
is at most
$20.00.
4.30 • p ≤
20.00

22. Example 3 Continued
4.30p ≤ 20.00
Since p is multiplied by 4.30, divide both sides by The symbol does not change.
p ≤ 4.65…
Since Jill can buy only whole numbers of tubes, she can buy 0, 1, 2, 3, or 4 tubes of paint.

23. daily cost at We Got Wheels
Example 3: Application
To rent a certain vehicle, Rent-A-Ride charges $55.00 per day with unlimited miles. The cost of renting a similar vehicle at We Got Wheels is $38.00 per day plus $0.20 per mile. For what number of miles is the cost at Rent-A-Ride less than the cost at We Got Wheels?
Let m represent the number of miles. The cost for Rent-A-Ride should be less than that of We Got Wheels.
Cost at Rent-A-Ride
must be less than
daily cost at We Got Wheels
plus
$0.20
per mile
times
# of miles. 55
< 38 +
0.20  m

24. Example 3 Continued
55 < m
Since 38 is added to 0.20m, subtract 38 from both sides to undo the addition.
–38 –38
55 < m
17 < 0.20m
Since m is multiplied by 0.20, divide both sides by 0.20 to undo the multiplication.
85 < m
Rent-A-Ride costs less when the number of miles is more than 85.

25. 1. Describe the solutions of 7 < x + 4.
Lesson Quiz: Part I
1. Describe the solutions of 7 < x + 4.
all real numbers greater than 3
2. Graph h ≥ –4.75 –5
–4.75
–4.5
Write the inequality shown by each graph.
x ≥ 3 3. 4.
x < –5.5