Warm-up:• Answer the following questions as accurately as you can:

1) What are 3 values that are less than 10? _____________

2) What are 3 values that are greater than –150? ________

3) How much money does a “rich” person have?

4) When someone has a fever, what temperature do they have?

5) How tall do you think you have to be to ride Drop Zone?

3-1 Graphing & Writing Inequalities

Compare. Write <, >, or =.1) 10 ____ 21 2) 5.27 ___ 5.23

3) 20% ____0.2 4) –4 ____ –5

< >

= >

An inequality is a……mathematical statement that compares two

quantities using one of the following signs:

< >

EXAMPLE 1: Applications with Inequalities

1A) Jimmy’s dad told him not to turn on the air conditioner unless the temperature was at least 82 degrees. Define a variable and write an inequality for the temperatures at which Jimmy can turn on the air conditioner. Graph the solutions.

82

Is 82 degrees included in the statement?

If it’s 82 degrees, should he turn the AC on? 81

If it’s 81 degrees, should he turn the AC on?

Yes, fill in the end point at 82.

83

If it’s 83 degrees, should he turn the AC on?

x = temperature

x 82

EXAMPLE 1: Applications with Inequalities

1B) The maximum speed allowed on the highway is 65 miles per hour. Define a variable and write an inequality for the situation. Graph the solutions.

65

Is 65 miles per hour included in the speed limit?

Are you allowed to go65 mph? 64

Are you allowed to go64 mph?

Yes. Fill in the end point at 65.

66x = speed (mph)

x 65

0

EXAMPLE 1: Applications with Inequalities

1C) It is recommended that the pool capacity at the community pool stay under 100 people. Define a variable and write an inequality for the situation. Graph the solutions.

100

Is 100 included in the possible number of people allowed in the pool?

If you were the 100th personto try to get in the pool, should they allow you in?

99Are 99 people allowed?

NO. Leave the end point empty.

101x = # of people

x 100

0(only whole #s)

The solution of an inequality is…… ANY value that makes the inequality true.

x 1213 14

1,000

12 13.1 13.2 12.99

Example:

Solutions:

• An inequality, such as x < 3, has too many solutions to list (an infinite # of solutions).

• One way to show all the solutions is to use a graph on a number line.

+–

3 42

Inequality Signs

<less than

>more than

When graphing:

less than equal to

more than

equal to

NOT

equal

Empty circle Empty circle Shaded circle Shaded circleEmpty circle

under over

Maximum, No morethan

At least,No less than

EXAMPLE 2: Graphing Inequalities2A)

x 5Rewrite w/variable on the

LEFTUse a straight edge: draw # line

Write the end point & put anempty or shaded circle.

5 64

Write a # on the left & rightof the endpoint

Use a line with an ARROW to show all solutions

Step 6: _____________________ Check a solution

If the variable is on the left, the

inequality sign will point in the direction

of the graph.

65

You read this as:“All real #s greater than or

equal to 5.”

ENDPOINT

+–

✔

EXAMPLE 2: Graphing Inequalities2B)

Rewrite with the variable on the LEFT

m 7Use a straight edge: draw # line

Write the end point & put anempty or shaded circle.

-7 -6-8Write a # on the left & right

of the endpointUse a line with an ARROW to

show all solutions

Step 6: _____________________ Check a solution

If the variable is on the left, the

inequality sign will point in the direction

of the graph.

7 m

8 7You read this as:

“All real #s less than –7.”

+–

✔

EXAMPLE 2: Graphing Inequalities2C)

Rewrite with the variable on the LEFT

d 3

4Use a straight edge: draw # line

Write the end point & put anempty or shaded circle. ¾ 1½

Write a # on the left & rightof the endpoint

Use a line with an ARROW to show all solutions

Step 6: _____________________ Check a solution

If the variable is on the left, the

inequality sign will point in the

direction of the graph.

3

4d

13

4

Practice sayingit with a classmate.

+–

✔

EXAMPLE 2: Graphing Inequalities2D)

Rewrite with right side simplified

x 9

Use a straight edge: draw # line

Write the end point & put anempty or shaded circle. 9 108

Write a # on the left & rightof the endpoint

Use a line with an ARROW to show all solutions

Step 6: _____________________ Check a solution

If the variable is on the left, the

inequality sign will point in the

direction of the graph.

x 42 (3 10)

89

+–

✔

EXAMPLE 3: Writing an Inequality from a graph

3A)11.511 12

Endpoint: 11.5How would we say what the graph shows?

“All real #s less than or equal to 11.5”

x

“Less than or equal to”

11.5

EXAMPLE 3: Writing an Inequality from a graph

3B)–4.2–4.3 –4.1

Endpoint: –4.2 How would we say what the graph shows?

“All real #s greater than –4.2”

x 4.2

EXAMPLE 3: Writing an Inequality from a graph

3C) 8988 90

Endpoint: 89How would we say what the graph shows?

“All real #s greater than or equal to 89”

x 89

Warm-up:Compare. Write <, >, or =.

1) 2)

3) 4)

5) Tell whether the inequality x < 5 is true or false for the following values of x:

• x = –10 b) x = 5 c) x = 4.99 d) x = –0.5

6) Solve:

15___ 10

6.5___6.05

0.25___1

4

7

8___

3

4

< >= <

True False True True

6x 8 2x 1 (3x 23)x = –2

Review: Writing Inequalities• Writing inequalities from words:“At least…” x ____ “No more than…” x _____

“At most…” x ____ “No less than…” x _____

“Maximum…” x ____ “Under…” x ______

Additional Practice with Word Problems:

• Define a variable and write an inequality for each situation. Graph the solutions.

1) You must be at least 52” to ride Drop Zone at Great America.

x = height

x 52 52 5351

Additional Practice with Word Problems:

• Define a variable and write an inequality for each situation. Graph the solutions.

2) The maximum number of pieces of Halloween candy I can eat before I feel sick is 15.

x = candy pieces

x 15 15 1614

0x 15