### Absolute Value Inequalities

Like absolute value equations, absolute value inequalities also will have two answers. However, they will have a range of answers, just like compound inequalities.

Consider the inequality

Notice in the second inequality, we did not write

Let's solve the following absolute value inequalities.

- Solve
|x+2|≤10.

There will be two solutions, one with the answer and sign unchanged and the other with the inequality sign flipped and the answer with the opposite sign.

Test a solution,

When graphing this inequality, we have

Notice that this particular absolute value inequality has a solution that is an “and” inequality because the solution is between two numbers.

If

If

If

If

If you are ever confused by the rules above, you can always test one or two solutions and graph it.

- Solve and graph
|4x−3|>9.

Break apart the absolute value inequality to find the two solutions.

Test a solution,

The graph is:

- Solve
|−2x+5|<11.

Given the rules above, this will become an “and” inequality, so the solution will be a range between two values.

The solution is

The graph is:

### Examples

#### Example 1

Earlier, you were asked to find the range of weights for a volleyball.

Set up an absolute value inequality where* w *is the range of weights of the volleyball.

So, the range of the weight of a volleyball is

#### Example 2

Is

Plug in -4 for

Yes, -4 works, so it is a solution to this absolute value inequality.

#### Example 3

Solve and graph

Split apart the inequality to find the two answers.

Test a solution,

### Review

Determine if the following numbers are solutions to the given absolute value inequalities.

|x−9|>4;10 ∣∣∣12x−5∣∣∣≤1;8 |5x+14|≥29;−8

Solve and graph the following absolute value inequalities.

|x+6|>12 |9−x|≤16 |2x−7|≥3 |8x−5|<27 ∣∣∣56x+1∣∣∣>6 |18−4x|≤2 - \begin{align*}\bigg | \frac{3}{4}x-8 \bigg |>13\end{align*}
- \begin{align*}|6-7x| \le 34\end{align*}
- \begin{align*}|19+3x| \ge 46\end{align*}

Solve the following absolute value inequalities. \begin{align*}a\end{align*} is greater than zero.

- \begin{align*}|x-a|>a\end{align*}
- \begin{align*}|x+a| \le a\end{align*}
- \begin{align*}|a-x| \le a\end{align*}

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 1.14.