Absolute Value Inequalities

G. O. L. A.

> Greater Thanis an“or”

Statement

Graphs go different directions.

Thumbs out<---|---|---|---|---|---|---|---|---|--->

< Less Thanis an“and”

Statement

Graphs must overlap Thumbs in

<---|---|---|---|---|---|---|---|---|--->

Remember, ‘and’ statements can be written as a compound

sentence X > -1 and X < 5

Tu

rn th

is on

ea

rou

nd

Brin

g th

is on

e d

ow

n

NOWPUT

THEM TOGETHER

1 x 5x

1 5x

Drop the

word “and”

1) | x + 4 | ≤ 2

Now what2 numberswill give us

this 2?

2 24x

Now find the value of “x”, set up 2 inequalities using the information found between the absolute value bars…..

One for the NEGATIVE

Don’t forget to flip the symbol

And one for the POSITIVE

DON’T FLIP!!!

4x 4 4x 6

4 4x 2

6 and 2x x

BUT FIRST – what happens to the inequality symbol when you multiply or divide by a negative number?

Nowput themtogether 6 2x

Let’s graph-6 ≤ x ≤-2

<---|---|---|---|---|---|---|---|---|---> -8 -7 -6 -5 -4 -3 -2 -1 0

Now shade

Between the

circles !!

What kind of

Circles?

When graphing COMPOUND INEQUALITIES

( the AND statements ), shading stops at the endpoints.

Another way to say it:

X is between -6 and -2

So….

Shade between -6 and -2

less than is an “and” statementless than is an “and” statementthumbs in to graphthumbs in to graph

What kind of statement?2) | x - 9 | ≤ 5

Now what 2 numberswill give us this 5?

5 59x 9x 9 9x 4

9 9x14

4 and 14x x Nowput themtogether 4 14x

Set up the NEGATIVE side

Don’t forget to flip the symbol

And set up the POSITIVE side

DON’T FLIP!!!

Let’s graph it.. 4 ≤ x ≤14

<---|---|---|---|---|---|---|---|---|---> 2 4 6 8 10 12 14 16 18

Now shade

Between the

circles !!

What kind of

Circles?

When graphing COMPOUND INEQUALITIES

( the AND statements ), shading stops at the endpoints.

Another way to say it:

X is between 4 and 14

So….

Shade between 4 and 14

3) | 4x – 5 | < 3 less than is an “and” statement – thumbs inless than is an “and” statement – thumbs in

What kind of statement?

3 34 5x 4 5x 5 5

4x 2

5 54x 8

1 and 2

2x x

Nowput themtogether

12

2x

Set up the NEGATIVE side

Don’t forget to flip the symbol

And set up the POSITIVE side

DON’T FLIP!!!

4 4 4 42

4x

1or

2x 2x

Graph it ½ < x < 2½ < x < 2

<---|---|---|---|---|---|---|---|---|---> 0 1 2 3

Now shade

Between the

circles !!

What kind of

Circles?

When graphing COMPOUND INEQUALITIES

( the AND statements ), shading stops at the endpoints.

Another way to say it:

X is between ½ and 2

So….

Shade between ½ and 2

4) | x | > 4

What kind of statement?

Graph

<---|---|---|---|---|---|---|---|---|--->

-4 -2 0 2 4

greater than is an “or” statementgreater than is an “or” statement

Remember, “or” in opposite directions – Thumbs outSet up the NEGATIVE side

Don’t forget to flip the symbol

And set up the POSITIVE side

DON’T FLIP!!!

4 4

4 or 4x x

x x

Or’sare not compound

statements!Don’t put them

together.

5) | 2x + 1 | ≥ 8

What kind of

statement?greater than is an “or” statement - Thumbs outgreater than is an “or” statement - Thumbs out

8 82 1x 2 1x 1 1

2x 91 1

2x 7

Set up the NEGATIVE side

Don’t forget to flip the symbol

And set up the POSITIVE side

DON’T FLIP!!!

2 2 2 29

2x

1or 4

2x 7

2x

1or 3

2x

1 14 or 3

2 2x x

<---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--->

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

1 14 or 3

2 2x x

6) | x + 9 | ≥ 7

What kind of statement?

7 79x 9x 9 9x 16

9 9x 2

16 or 2x x

Set up the NEGATIVE side

Don’t forget to flip the symbol

And set up the POSITIVE side

DON’T FLIP!!!

<---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|--->

-16 -14 -12 -10 -8 -6 -4 -2 0 2

7) | x + 9 | ≥ -5

The absolute value of any number is positive, therefore

| x + 9 | ≥ -5

+ ≥ -

Absolute

ValueIs

Pos.

A POSITIVE IS ALWAYS

GREATER THAN

A NEGATIVE

Therefore

ALL REAL NUMBERS

Is the answer!!!

8) | x - 6 | ≤ -2

The absolute value of any number is positive, therefore

| x - 6 | ≤ -2

+ ≤ -

Absolute

ValueIs

Pos.

A POSITIVE IS NEVER

LESS THAN

A NEGATIVE

Therefore

NO SOLUTION

Is the answer!!!