Absolute Value Equations

Absolute value measures distance from zero on a number line. Since distance is always non-negative, absolute value strips away the sign of a number. This simple idea leads to equations with a surprising twist: they can produce two solutions.

Review: What Is Absolute Value?

|x| = x if x \geq 0, and |x| = -x if x < 0

In simpler terms: |5| = 5 and |-5| = 5. Both 5 and -5 are a distance of 5 from zero.

The Key Principle

If |x| = a, where a > 0, then:

x = a or x = -a

There are always two solutions (when a > 0), because two numbers on the number line have the same distance from zero.

Worked Example 1: Simple Absolute Value

Solve: |x| = 7

Apply the principle: x = 7 or x = -7
Check: |7| = 7 and |-7| = 7. Both work.

Solving |expression| = value

When the absolute value contains an expression, set up two equations:

Worked Example 2: Expression Inside

Solve: |2x - 3| = 11

Set up two equations: 2x - 3 = 11 or 2x - 3 = -11
Solve the first: 2x = 14, so x = 7
Solve the second: 2x = -8, so x = -4
Check: |2(7) - 3| = |11| = 11. |2(-4) - 3| = |-11| = 11. Both correct.

Worked Example 3: Isolate First

Solve: |3x + 1| + 5 = 12

Isolate the absolute value: |3x + 1| = 12 - 5 = 7
Set up: 3x + 1 = 7 or 3x + 1 = -7
First equation: 3x = 6, x = 2
Second equation: 3x = -8, x = -8/3
Check both in the original equation to verify.

Special Cases: No Solution

Absolute Value Cannot Be Negative

If you isolate the absolute value and get |expression| = negative number, the equation has no solution. For example, |x + 2| = -3 has no solution because absolute value is never negative.

One more special case: |expression| = 0 has exactly one solution, because the only number with absolute value 0 is 0 itself.

Worked Example 4: No-Solution Case

Solve: |4x - 1| + 6 = 2

Isolate: |4x - 1| = 2 - 6 = -4
An absolute value cannot equal -4.
No solution.

The Distance Interpretation

|x - 5| = 3 means "the distance between x and 5 is 3." On the number line, x is 3 units away from 5, so x = 8 or x = 2. This geometric view can help you set up equations quickly.

Practice Problems

1. Solve: |x| = 12

Show Solution

x = 12 or x = -12

2. Solve: |x + 4| = 9

Show Solution

x + 4 = 9 gives x = 5. x + 4 = -9 gives x = -13. Solutions: x = 5 or x = -13.

3. Solve: |5x - 10| = 0

Show Solution

5x - 10 = 0, so 5x = 10, x = 2. One solution: x = 2.

4. Solve: 2|x - 3| + 1 = 15

Show Solution

Isolate: 2|x - 3| = 14, so |x - 3| = 7. Then x - 3 = 7 gives x = 10, and x - 3 = -7 gives x = -4. Solutions: x = 10 or x = -4.

5. Solve: |2x + 6| = -5

Show Solution

Absolute value cannot be negative. No solution.

Lesson Summary

|expression| = a (where a > 0) gives two equations: expression = a or expression = -a.
Always isolate the absolute value expression before splitting into two cases.
If |expression| = negative, there is no solution.
If |expression| = 0, there is exactly one solution.
Absolute value represents distance from zero (or distance between values).