4-6 Solving Absolute Value Equations & Inequalities

Absolute Value (of x)Symbol Represents the distance x is from zero on the

number line.Always positiveEx:

-4 -3 -2 -1 0 1 2

x

33

Ex: What are the possible values of x?

x = 5 or x = -5

So there are two answers!

5x

Solving an absolute value equation:ax+b = c, where c>0

This problem will also have two answers!Set up 2 new equations, then solve.

ax+b = c and ax+b = -c

NB: Make sure the absolute value is by itself before you split to solve

If the absolute value isn’t by itself, solve until it is.

Ex.

Don’t set up 2 new equations until the absolute value is by itself!

cbax isn’t the same as cbax

8372 x

Ex: To solve set up two equations…

6x-3 = 15 or 6x-3 = -15

1536 x

Add 3 to both sides:

6x = 18 or 6x = -12Divide both sides by 6

x = 3 or x = -2

* Plug in answers to check your solutions!

Ex: To solveGet the absolute value part by itself by adding 3 to

both sides!

8372 x

8372 x3 3

1172 xNow split into 2 parts.

2x+7 = 11 or 2x+7 = -11Add (-7) to both sides2x = 4 or 2x = -18

Divide both sides by 2x = 2 or x = -9

Check the solutions.

You can also solve: Absolute Value Inequalities!

The method for solving Absolute Value Inequalities depends on the inequality symbol.

Think about what this means…what values for x make this statement true?Let’s plot them on a number line.

3x

If it’s a less than problem, change it into an “and” compound inequality.

means AND

Another way to write this is…..

3x 3x 3x

33 x

The method for solving Absolute Value Inequalities depends on the inequality symbol.

If it’s a less than problem, change it into an “and” compound inequality.

33 x 3x

To write it algebraically:

cbax cbaxc

If it’s a greater than problem, change it into an “or” compound inequality.

The method for solving Absolute Value Inequalities depends on the inequality symbol.

3xThink about what this means…what values for x make this statement true?Let’s plot them on a number line.

If it’s a greater than problem, change it into an “or” compound inequality.

means OR 3x 3x 3x

The method for solving Absolute Value Inequalities depends on the inequality symbol.

If it’s a greater than problem, change it into an “or” compound inequality.

3x 3x

To write it algebraically:

OR 3x

Becomes an “or” problem

Change to: ax+b > c or ax+b < -c

cbax

Solve & graph.

Less than means it becomes an “and” problem

732 x

7327 x* Add 3

1024 x* Divide by 2

52 x

Solve & graph.

Get absolute value by itself first.822 x

822or 822 xx

62or 102 xx

3or 5 xx

11322 x

Add -3

Add 2

Divide by 2

Since it’s greater than, it becomes an “or” problem