Here are the steps to follow when solving absolute valueinequalities:

- Isolate the absolute value expression on the left side ofthe inequality.
- If the number on the other side of the inequality sign isnegative, your equation either has no solution or all real numbers as solutions.Use the sign of each side of your inequality to decide which of these casesholds. If the number on the other side of the inequality sign is positive,proceed to step 3.
- Remove the absolute value bars by setting up a compound inequality.The type of inequality sign in the problem will tell us how to set up thecompound inequality. If your problem has a
- Solve the inequalities.

**greater than**sign (your problemnow says that an absolute value is greater than a number), then set upan "or" compound inequality that looks like this:

(quantity inside absolute value) < -(number on otherside)

OR

(quantity inside absolute value) > (number on other side)

The same setup is used for a ³sign.

If your absolute value is **less than** a number, thenset up a three-part compound inequality that looks like this:

-(number on other side) < (quantity inside absolutevalue) < (number on other side)

The same setup is used for a £sign

This process can be a little confusing at first, so bepatient while learning how to do these problems. Lets look at some examples.

**Example 1: |x + 4| - 6 < 9**

Step 1: Isolate the absolutevalue | |x + 4| < 15 |

Step 2: Is the number onthe other side negative? | No, its a positive number, 15.Well move on to step 3. |

Step 3: Setup a compound inequality | -15 < x + 4 < 15 |

Step 4: Solvethe compound inequality |

**Example 2: |2x 1| - 7 ³-3**

Step 1: Isolate the absolutevalue | |2x 1| ³ 4 |

Step 2: Isthe number on the other side a negative number? | No, its a positive number, 4.Well move on to step 3. |

Step 3: Setup a compound inequality | 2x 1 £ -4 or 2x 1³ 4 |

Step 4: Solvethe inequalities | 2x £ -3 or 2x ³5 x £ -3/2 or x ³5/2 |

**Example 3: |5x + 6| + 4 < 1**

Step 1: Isolatethe absolute value | |5x + 6| < -3 |

Step 2: Isthe number on the other side a negative number? | Yes, its a negative number, -3. Well look at the signs of each side of the inequalityto determine the solution to the problem: |5x + 6| < -3 positive < negative This statement is never true, so there is no solutionto this problem. |

**Example 4: |3x 4| + 9 > 5**

Step 1: Isolatethe absolute value | |3x 4| > -4 |

Step 2: Isthe number on the other side a negative number? | Yes, its a negative number, -4. Well look at the signs of each side of the inequalityto determine the solution to the problem: |3x 4| > -4 positive > negative This statement is always true, so the solution to theproblem is All Real Numbers |