When an inequality has an absolute value we will have to remove the absolute value in order to graph the solution or give interval notation. The way we remove the absolute value depends on the direction of the inequality symbol.
Consider .
Absolute value is defined as distance from zero. Another way to read this inequality would be the distance from zero is less than 2. So on a number line we will shade all points that are less than 2 units away from zero.
This graph looks just like the graphs of the three part compound inequalities! When the absolute value is less than a number we will remove the absolute value by changing the problem to a three part inequality, with the negative value on the left and the positive value on the right. So becomes , as the graph above illustrates.
Consider .
Absolute value is defined as distance from zero. Another way to read this inequality would be the distance from zero is greater than 2. So on the number line we shade all points that are more than 2 units away from zero.
This graph looks just like the graphs of the OR compound inequalities! When the absolute value is greater than a number we will remove the absolute value by changing the problem to an OR inequality, the first inequality looking just like the problem with no absolute value, the second flipping the inequality symbol and changing the value to a negative. So becomes or , as the graph above illustrates.
We can solve absolute value inequalities much like we solved absolute value equations by following these steps.
EXAMPLE
Solve, graph, and give interval notation for the solution

Absolute value is greater, use OR  

Solve  

Add 5 to both sides  

Divide both sides by 4  




Graph 
Interval notation 
EXAMPLE
Solve, graph, and give interval notation for the solution

Add 4 to both sides to isolate the absolute value  




Divide both sides by  

Dividing by a negative switches the symbol  




Absolute value is greater, use OR; Graph 
Interval notation 
EXAMPLE
Solve, graph, and give interval notation for the solution

Subtract 9 from both sides  




Divide both sides by  

Dividing by negative switches the symbol  




Absolute value is less, use three part; Solve  

Subtract 1 from all three parts  

Divide all three parts by 4  




Graph 
Interval Notation 
It is important to remember as we are solving these equations, the absolute value is always positive. There are cases where there is no solution to the inequality or all real numbers are the solution:

Subtract 12 from both sides  




Divide both sides by 4  




Absolute value can't be less than a negative 
Interval Notation 
Solve, graph, and give interval notation for the solution

Subtract 5 from both sides  




Divide both sides by  

Dividing by a negative flips the symbol  

Absolute value always greater than negative 
Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html