Absolute value inequalities and equations are a bit tricky to work with. There are basically three cases or situations that can arise when working with them. By guessing and checking we can answer the following three questions.

**Tip**: We can easily generalize the above result so that we can use this idea as a template when solving equations and inequalities with absolute values in them. ( Assume

*n*> 0 )

**Case 1**: |

*X*| =

*n*can be solved using

*X*= -

*n*or

*X*=

*n*.

**Case 2**: |

*X*| <

*n*can be solved using -

*n*<

*X*<

*n.*

**Case 3**: |

*X*| >

*n*can be solved using

*X*< -

*n*or

*X*>

*n*.

Use the following steps to solve an absolute value equation or inequality.

**Step 1**: Isolate the absolute value.

**Step 2**: Identify the case and apply the appropriate theorem.

**Step 3**: Solve the resulting equation or inequality.

**Step 4**: Graph the solution set and express it in interval notation.

**Instructional Video**: Absolute Value Inequalities

**Solve and graph the solution set**.

In the three cases listed above notice that we took care to say that

*n*> 0. The next three problems illustrate some of the situations encountered when

*n*= 0. Plug in some numbers and see what happens.

**YouTube Videos**: