Absolute Value Inequalities

Here are the steps to follow when solving absolute value inequalities:

1. Isolate the absolute value expression on the left side of the inequality.

2. If the number on the other side of the inequality sign is negative, 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 cases holds. If the number on the other side of the inequality sign is positive, proceed to step 3.

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 the compound inequality.

4. If your problem has a greater than sign (your problem now says that an absolute value is greater than a number), then set up an "or" compound inequality that looks like this:

(quantity inside absolute value) < -(number on other side)
OR
(quantity inside absolute value) > (number on other side)

The same setup is used for a ³ sign.

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

-(number on other side) < (quantity inside absolute value) < (number on other side)

The same setup is used for a £ sign

5. Solve the inequalities.

This process can be a little confusing at first, so be patient while learning how to do these problems. Let’s look at some examples.

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

 Step 1: Isolate the absolute value |x + 4| - 6 < 9 |x + 4| < 15 Step 2: Is the number on the other side negative? No, it’s a positive number, 15. We’ll move on to step 3. Step 3: Set up a compound inequality The inequality sign in our problem is a less than sign, so we will set up a 3-part inequality: -15 < x + 4 < 15 Step 4: Solve the compound inequality -19 < x < 11

Example 2: |2x – 1| - 7 ³ -3

 Step 1: Isolate the absolute value |2x – 1| - 7 ³ -3 |2x – 1| ³ 4 Step 2: Is the number on the other side a negative number? No, it’s a positive number, 4. We’ll move on to step 3. Step 3: Set up a compound inequality The inequality sign in our problem is a greater than or equal to sign, so we will set up a compound inequality with the word "or": 2x – 1 £ -4 or 2x – 1 ³ 4 Step 4: Solve the inequalities 2x – 1 £ -4 or 2x – 1 ³ 4 2x £ -3 or 2x ³ 5 x £ -3/2 or x ³ 5/2

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

 Step 1: Isolate the absolute value |5x + 6| + 4 < 1 |5x + 6| < -3 Step 2: Is the number on the other side a negative number? Yes, it’s a negative number, -3.  We’ll look at the signs of each side of the inequality to determine the solution to the problem: |5x + 6| < -3 positive < negative This statement is never true, so there is no solution to this problem.

Example 4: |3x – 4| + 9 > 5

 Step 1: Isolate the absolute value |3x – 4| + 9 > 5 |3x – 4| > -4 Step 2: Is the number on the other side a negative number? Yes, it’s a negative number, -4.  We’ll look at the signs of each side of the inequality to determine the solution to the problem: |3x – 4| > -4 positive > negative This statement is always true, so the solution to the problem is All Real Numbers