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 zero.

figure

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

| 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.

figure

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

{ x| x < –4 or x > 4}

| 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

  • | ax + b| < c or | ax + b| > c

  • | ax + b| ≤ c or | ax + b| ≥ c

  • If c is negative,

    • | ax + b| < c has no solutions.

    • | ax + b| ≤ c has no solutions.

    • | ax + b| > c has as its solution all real numbers.

    • | ax + b| ≥ c has as its solution all real numbers.

  • If c = 0,

    • | ax + b| < 0 has no solutions.

    • | ax + b| ≤ 0 has as its solution the solution to ax + b = 0.

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

    • | ax + b| ≥ 0 has as its solution all real numbers.

  • If c is positive,

    • | ax + b| < c has solutions that solve

      ax + b > – c and ax + b < cc < ax + b < c

      That is:

      • | ax + b| > c has solutions that solve

        ax + b < – c or ax + b > c

      • | ax + b| ≤ c has solutions that solve

        cax + bc

      • | ax + b| ≥ c has solutions that solve

        ax + b ≤ – c or ax + bc

Example 1

Solve for x: |3 x – 5| < 12.

equation

The solution set is equation

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

Figure 3. x is greater than equation and less than equation.

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Example 2

Solve this disjunction for x: |5 x + 3| > 2.

equation

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

Figure 4. x is less than –1 or greater than equation.

figure

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, equation

The solution of the set is equation. The graph of the solution set is shown in Figure 5.

Figure 5. All numbers except equation.

figure

Example 5

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

First, isolate the e xpression involving the absolute value symbol.

equation

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.

figure