Absolute Value Inequalities

When solving an absolute value equation or inequality, understand that the value of the “inside” of the absolute value symbol may be positive or negative ... once the absolute value is evaluated the result is always positive.  In other words,  or .

Absolute value inequalities are solved in a similar fashion as absolute value equations.  Understand that the “less than” and “less than or equal to” inequalities are compound statements.

Solve the inequality: .   This would mean that the value for  would lie between the values of  5  and  –5.  Therefore, you would set up a compound inequality to solve the problem.
The problem would now look like this:        

Solving this compound inequality would provide the result that .  Which means that the value for x lies between  –1  and  4. 

(graph)                               [             ]                                                         

                                         -1               4

(interval notation)                        … notice that this is an Intersection of two sets of numbers.

The “greater than” and “greater than or equal to” inequalities are disjoint statements.  This means that you must “dis”join them or take them apart to solve.  The answer to the inequality will be greater than the positive value of the number “or” less than the negative value of the number.  The answer is the union of the two inequalities.

Solve the inequality: .   This would mean that the value for  would be greater than  5  or the value for  would be less than  –5.  When you “dis”join the inequality to solve, the problem would look like this:           “or” 

Solving these inequalities would provide the result  “or” .                                          

(graph)                                       )            (      

                                                 -1            4

(interval notation)               … notice that this is a Union of two sets of numbers.

General Algebra Tips

The views and opinions expressed in this page are strictly those of Mary Lou Baker.
The contents of this page have not been reviewed or approved by Columbia State Community College.

This page was edited on 09-Jan-2014