Absolute Value Equations

all that the absolute value of a number represents the distance that number is from zero on the number line. The equation | x| = 3 is translated as “ x is 3 units from zero on the number line.” Notice, on the number line shown in Figure 1, that two different numbers are 3 units away from zero, namely, 3 and –3.

Figure 1. Absolute value.

figure Absolute value.

The solution set of the equation | x| = 3 is {3, –3}, because |3| = 3 and |–3| = 3.

Example 1

Solve for x: |4 x – 2| = 8.

This translates to “4 x – 2 is 8 units from zero on the number line” (see Figure 2).

equation

Check the solution.

equation

These are true statements. The solution set is equation.

Figure 2. There are + and – solutions.

figure

Example 2

Solve for x:

equation

To solve this type of absolute value equation, first isolate the expression involving the absolute value symbol.

equation

Now, translate the absolute value equation: “ equation is 11 units from zero on the number line.”

equation

The check is left to you. The solution set is equation.

Example 3

Solve for x: | x | = –2.

This problem has no solutions, because the translation is nonsensical. Distance is not measured in negative values.

Example 4

Solve for x: |2 x – 3| = |3 x + 7|.

This type of sentence will be true if either

  1. The expressions inside the absolute value symbols are exactly the same (that is, they are equal); or

  2. The expressions inside the absolute value symbols are opposites of each other.

equation

The check is left to you. The solution set is equation.

Example 5

Solve for x: | x – 2| = |7 – x|.

equation

The sentence –2 = –7 is never true, so it gives no solution. So the only possible solution is equation.

Check the solution.

equation

Therefore, the solution set is equation.