Inequalities with absolute values are solved by first getting the absolute value by itself, then dropping the absolute value and solving for the positive and negative cases. This may seem confusing and is best understood by working through a few examples.

 

Example

 

Solve ½x ½> 3

 

The absolute value is already by itself in that side of the inequality, so the next step is to drop the absolute value and look at the positive and negative cases. When looking at the negative case, flip the sign around.

 

x > 3 and x < -3

 

Notice that the negative case, x < -3, has the sign flipped around. When looking at the negative case make sure that the entire side is multiplied by a negative and not just the first term. This problem only has 1 term, -3, so that is not an issue.

 

-3 > x > 3

 

Answer!

 

Note- There is not a number that will meet both conditions, being less than –3 and greater than 3. Thus, this would be referred to as an “or” statement.

 

Example

 

Solve 2½x - 4 ½£ 10

 

Get the absolute value by itself by dividing by 2.

 

½x - 4 ½£ 5

 

Drop the absolute value and look at the positive and negative cases. Make sure the sign is flipped around when looking at the negative case.

 

x – 4 £ 5 and

x – 4 ³ -5

 

Add 4 to both sides of both inequalities to get x by itself.

 

x £ 9 and

x ³ -1

 

Since there are numbers that can meet both conditions, this is referred to as an “and” statement.

 

 

- 1 £ x £ 9

 

Answer!