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.

**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.

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