Absolute Value — What it Means

The absolute value of a number x, written |x|, means "how far x is from 0" on the number line. Distance is always not negative, so |x| is never less than 0.

Simple definition

|x| = x if x is zero or positive. |x| = -x if x is negative. Example: |5| = 5, |-5| = 5, |0| = 0.

Number-line idea (visual)

-5 -4 -3 -2 -1  0  1  2  3  4  5
        ^         ^
      |-3| = 3    |3| = 3

Both -3 and 3 are three units away from 0, so both have absolute value 3.

How to solve equations with absolute value

1) If |x| = a where a >= 0, then x = a or x = -a. (Because two numbers can be the same distance from 0.)

Example: |x| = 4 → x = 4 or x = -4.

2) If |x| = 0, then x = 0.

How to solve inequalities with absolute value

1) |x| < a (where a > 0): means x is closer than a units to 0. So -a < x < a.

Example: |x| < 3 → -3 < x < 3.

2) |x| > a (where a > 0): means x is farther than a units from 0. So x < -a or x > a.

Example: |x| > 2 → x < -2 or x > 2.

Some useful properties

• |ab| = |a| |b|
• |a/b| = |a| / |b| if b ≠ 0
• |a + b| ≤ |a| + |b| (triangle inequality: the distance from 0 of a+b is at most the sum of distances)

Quick practice (try these)

• Compute: | -8 | → answer: 8
• Solve: |x| = 6 → answers: x = 6 or x = -6
• Solve: |x - 2| < 3 → move 2: -3 < x - 2 < 3 → add 2: -1 < x < 5
• Solve: |x + 1| > 4 → x + 1 < -4 or x + 1 > 4 → x < -5 or x > 3

Common mistakes and tips

• Remember |x| is never negative. If someone writes |x| = -3, that has no solution.
• When solving |expression| < a, convert to a compound inequality: -a < expression < a.
• When solving |expression| > a, split into two separate inequalities: expression < -a or expression > a.
• Think of absolute value as distance — this often makes inequalities easier to understand.

If you want, try a few more problems and I can check your answers or show step-by-step solutions.