How to Solve Absolute Value Inequalities

Equations with absolute values and inequalities have the form

|ax+b| > c    or    |ax+b| < c

where x is the variable in the algebra problem, and the parameters a, b, and c are constants, and c is positive. These problems are frequently encountered on standardized tests such as the SAT, ACT, GRE, and GMAT. They also arise in many practical applications, such as determining distances between points.

When you solve algebraic inequalities, the solution for x is actually a range of values. And when you solve math problems involving absolute values, you must always analyze two cases. This makes absolute value inequality problems rather challenging because the solution is given in terms of two inequalities. You can use the outline below as a guide when solving these types of algebra problems, or use the absolute value inequality calculator on the left.

Step 1

If the inequality is in the form |ax+b| > c or |ax+b| ≥ c, you must set up and solve these two equations:

ax+b >/≥ c    and    ax+b </≤ -c.

If the inequality is in the form |ax+b| < c or |ax+b| ≤ c, you must set up and solve these two equations:

ax+b </≤ c    and    ax+b >/≥ -c.

These are now regular algebra equations in one variable.

Step 2

Use the rules of inequalities to solve both equations for the range of x values. Remember, if you divide or multiply both sides of an equality by a negative number, you must switch the direction of the inequality. Your final result will be two inequalities that represent the possible values for x.

Example

Solve the equation |2x-9| > 13. The first step is to split it into two equations:

2x - 9 > 13    and    2x - 9 < -13

The first inequality equation can be simplified to 2x > 22, or x > 11. The second equation can be simplified to 2x < -4, or x < -2. So the full solution is {x > 11, x < -2}. This range of solutions happens to be a disjoint set. Another way of expressing the solution is that x can equal all real numbers except for numbers between -2 and 11 (inclusive).

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