# Solving absolute value inequalities

When solving absolute value inequalities, the process is very similar to solving absolute value equations. You should review the latter before studying this lesson.

## Solving absolute value inequalities using the definition of absolute value.

Absolute value definition:

If x is positive, | x | = x

If x is negative, | x | = -x

Example #1:

Solve for x when | x | < 8

After applying the definition to example #1, you will have two equations to solve.

In fact, when solving absolute value inequalities, you will usually get two solutions. That is important to keep in mind!

If x is positive, | x | = x, so the first equation to solve is x < 8. This is done since x is automatically isolated.

If x is negative, | x | = -x, so the second equation to solve is -x < 8.

You can write -x < 8 as -1x < 8 and divide both sides by -1 to isolate x.

(-1/-1)x > 8/-1

1x > 8/-1

x > -8

Notice that the smaller sign (<) was switched to a bigger sign (>). This happens whenever you divide or multiply inequalities by a negative number.

Let us see why this makes sense!

We know that 2 < 4. However, let us divide both sides by a negative number, say -2

2/-2 ? 4/-2

-1 ? -2

Should ? be < or >?

Since -1 is bigger than -2, it should be -1 > -2

So, the solutions are x < 8 and x > -8

x > -8 means the same thing as -8 < x

Putting -8 < x and x < 8 together, you can write -8 < x < 8

Therefore, any number between -8 and 8 is a solution.

for instance, if we choose -5, we get | -5 | = 5 and 5 is smaller than 8.

Example #2:

Solve for x when | x − 4 | < 7

Before, we apply the definition, let's make a useful substitution

Let y = x − 4, so | x − 4 | < 7 becomes | y | < 7. You must understand this step. No excuses!

Now, let's apply the definition to | y | < 7. Again, you will have two inequalities to solve

Once again, when solving absolute value inequalities, you will usually get two solutions.

If y is positive, | y | = y, so the first equation to solve is y < 7. No, you are not done! You have to substitute x − 4 for y

After substitution, y < 7 becomes x − 4 < 7

x − 4 < 7

x + -4 < 7

x + -4 + 4 < 7 + 4

x < 11

If y is negative, | y | = -y, so the second equation to solve is -y < 7.

You have to substitute x − 4 for y

You get -( x − 4) < 7. Notice the inclusion of parenthesis this time

-(x − 4) < 7

-(x + -4) < 7

Multiplying both sides by -1

-1 × -(x + -4) > -1 × 7 (Change < to >)

x + -4 > -7

x + -4 + 4 > -7 + 4

x > -3

The solutions are x > -3 and x < 11

This is equivalent to -3 < x < 11

Example #3:

Solve for x when | 3x + 3 | > 15

Before, we apply the definition, let's make a useful substitution.

Let y = 3x + 3, so | 3x + 3 | > 15 becomes | y | > 15.

Now, let's apply the definition to | y | > 15.

Lastly, when solving absolute value inequalities, you will usually get two solutions. We may never say this enough!

If y is positive, | y | = y, so the first equation to solve is y > 15. You have to substitute 3x + 3 for y

After substitution, y > 7 becomes 3x + 3 > 15

3x + 3 > 15

3x + 3 − 3 > 15 − 3

3x > 12

(3/3)x > 12/3

x > 4

If y is negative, | y | = -y, so the second equation to solve is -y > 15.

After substitution, -y > 7 becomes -(3x + 3) > 15

-(3x + 3) > 15

-3x + -3 > 15

-3x + -3 + 3 > 15 + 3

-3x > 18

(-3/-3)x > 18/-3

x < -6

The solutions are x > 4 and x < -6

Solving absolute value inequalities should be straightforward if you follow my guidelines above

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If you can solve these problems with no help, you must be a genius!

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