Inequalities and Their Graphs

Inequalities –

What do they mean in words?•Less than or smaller than

•Fewer than

•Less than or equal to

•At most

•No more than

•A maximum of

•Greater than or bigger than

•More than

•Greater than or equal to

•At least

•No less than

•A minimum of

Example:

Write an inequality to describe people who are NOT of legal driving age in California

First- try to put it in words

Who are the people who can not drive?

Everyone younger than or less than 16 years old

people who can’t drive are less than 16

p<16

When we start graphing we have to remember…

Use OPEN bullet for

Use CLOSED bullet for

1. Graph X -1

2. Graph K >

3. Write the inequality

-1 1 20 3-2

-1 1 20 3-2

-1 1 20 3-2

X 2

Try this one on your own or with your partner

Solve AND graph it yy

yy

-7y -7y

4 > y - 10

+10 +10

14 > y 13 15 1614 1712

Check out what happens with Division AND Multiplication

12 > 4 12 > 4THIS IS TRUE THIS IS TRUE

44 -4 -4

3 > 1THIS IS STILL TRUE

-3 > -1NOT TRUE

So something weird happens when we divide by a negative

THE RULES

If you divide OR multiply by a negative number you must “flip”

the sign

>

Example

€

−3x ≤ 27

We divided by a negative we have to

FLIP

€

−3

€

−3

x -9

€

≥

Solve & Graph each Inequality

€

4(x − 3) + 7 ≥ 4x +1

No Solution – It doesn’t make sense!

Solve & Graph each Inequality

€

2x < 2(x +1) + 3

ALL real numbers are solutions!

Writing Compound InequalitiesWriting Compound Inequalities

x 4

l l l l l l l

-6 -5 -4 -3 -2 -1 0

x is greater than –4 and less than or equal to –2.

and x 2 4 x 4 2x

Writing Compound InequalitiesWriting Compound Inequalities

x 3

l l l l l l l

-2 -1 0 1 2 3 4

x is greater than 3 or less than –1.

or x 1

Solving a Compound Inequality with AndSolving a Compound Inequality with AndSolve the inequality and graph the solution.

1 1 4

62 2 2 3

l l l l l l l l

-3 -2 -1 0 1 2 3

12x

2 x€

2x +1

€

−3

€

7

Solving a Compound Inequality with AndSolving a Compound Inequality with AndSolve the inequality and graph the solution.

4 6 3 4. x3 3

3 1

l l l l l l l l

-5 -4 -3 -2 -1 0 1

3x

Solving a Compound Inequality with AndSolving a Compound Inequality with And

Solve the inequality and graph the solution.

5 2 3 8 17. x 8 8

6 9

l l l l l l l l

-4 -3 -2 -1 0 1 2

8 3x

3 32 3

3 x

3 2x

Solving a Compound Inequality with OrSolving a Compound Inequality with Or

Solve the inequality and graph the solution.

6 2 3 5 3 1 16. x or x 32x

l l l l l l l l

3 4 5 6

38

2 2x 4

13x

115

3 3x 5or

Solving a Compound Inequality with OrSolving a Compound Inequality with OrSolve the inequality and graph the solution.

7 2 6 2 6. 4 x or x 2

4x

l l l l l l l l

-6 -5 -4 -3 -2 -1 0

24

4 4x 1

2 2

x 3or

Writing and Using a Linear ModelWriting and Using a Linear ModelIn 1985, a real estate property was sold for $172,000. The property was sold again in 1999 for $226,000. Write a compound inequality that represents the different values that the property was worth between 1995 and 1999.x 172 000, x 226 000,

172 000 226 000, , xand172 000, x