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.

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). 

Check the solution.

These are true statements. The solution set is

Figure 2. There are + and – solutions.

Example 2

Solve for x:

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

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

The check is left to you. The solution set is

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. 

The check is left to you. The solution set is

Example 5

Solve for x: | x – 2| = |7 – x|. 

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

Check the solution.

Therefore, the solution set is

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