Absolute Value Formula

The Formula

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

When to use: -5 and 5 are both 5 units from zero, so |-5| = |5| = 5.

Quick Example

|-7| = 7 (distance 7 from zero), |3| = 3, |0| = 0; also |5 - 8| = 3.

Notation

|x| means the absolute value of x

What This Formula Means

The distance of a number from zero on the number line, always non-negative; written |x|.

-5 and 5 are both 5 units from zero, so |-5| = |5| = 5.

Formal View

|x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}, \quad \text{equivalently } |x| = \sqrt{x^2}

Worked Examples

Example 1

easy
Evaluate |{-7}| + |3|.

Solution

  1. 1
    The absolute value of -7 is the distance from 0: |{-7}| = 7.
  2. 2
    The absolute value of 3 is |3| = 3.
  3. 3
    Add: 7 + 3 = 10.

Answer

10
Absolute value measures distance from zero on the number line and is always non-negative. For any number a, |a| = a if a \ge 0 and |a| = -a if a < 0.

Example 2

medium
Evaluate |5 - 12| - |2 - 9|.

Common Mistakes

  • Thinking |-a| = -a
  • Not distributing absolute value correctly

Why This Formula Matters

Used for distances, errors, and tolerances where direction doesn't matter.

Frequently Asked Questions

What is the Absolute Value formula?

The distance of a number from zero on the number line, always non-negative; written |x|.

How do you use the Absolute Value formula?

-5 and 5 are both 5 units from zero, so |-5| = |5| = 5.

What do the symbols mean in the Absolute Value formula?

|x| means the absolute value of x

Why is the Absolute Value formula important in Math?

Used for distances, errors, and tolerances where direction doesn't matter.

What do students get wrong about Absolute Value?

Confusing |-x| with -|x|: |-3| = 3 but -|{-3}| = -3. Always non-negative inside.

What should I learn before the Absolute Value formula?

Before studying the Absolute Value formula, you should understand: integers.

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