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Absolute Value Inequalities

Remember, absolute value means distance from zero on a number line. | x| < 4 means that x is a number that is less than 4 units from zero on a number line (see Figure 1 ).





Figure 1

Less than 4 from 0.

The solutions are the numbers to the right of –4 and to the left of 4 and could be indicated as




| x| < 4 means that x is a number that is more than 4 units from zero on a number line (see Figure 2 ).





Figure 2

More than 4 from 0.

The solutions are the numbers to the left of –4 or to the right of 4 and are indicated as




| x| > 0 has no solutions, whereas | x| > 0 has as its solution all real numbers except 0. | x| > −1 has as its solution all real numbers, because after taking the absolute value of any number, that answer is either zero or positive and will always be greater than −1.

The following is a general approach for solving absolute value inequalities of the form




  • If c is negative, has no solutions. has no solutions. has as its solution all real numbers. has as its solution all real numbers.

  • If c = 0, has no solutions. has as its solution the solution to ax + b = 0.

    has as its solution all real numbers, except the solution to ax + b = 0. has as its solution All real numbers.

  • If c is positive, has solutions that solve




    That is, has solutions that solve



Example 1: Solve for




The solution set is



The graph of the solution set is shown in Figure 3 .





Figure 3

x is greater than − 73 and less than 173


Example 2: Solve for x | 5 x + 3| > 2




The solution set is . The graph of the solution set is shown in Figure 4 .





Figure 4

x is less than –1 or greater than −1/5


Example 3: Solve for x: |2 x + 11| < 0

There is no solution for this inequality.

Example 4: Solve for x: |2 x + 11| > 0

The solution is all real numbers except for the solution to 2 x + 11 = 0. Therefore,




The solution of the set is is a real number, . The graph of the solution set is shown in Figure 5 .





Figure 5

All numbers but − 112

Example 5: Solve for x: 7 |3 x + 2| + 5> 4

First, isolate the expression involving the absolute value symbol.




The solution set is all real numbers. ( Note: The absolute value of any number is always zero or a positive value. Therefore, the absolute value of any number is always greater than a negative value.) The graph of the solution set is shown in Figure 6 .





Figure 6

The set of all numbers.

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