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|. For any real number, absolute value strips away the sign and returns only the magnitude.

-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| always equals -a — it equals the positive magnitude regardless of input sign
  • Incorrectly distributing absolute value across operations — |a + b| \neq |a| + |b| in general
  • Forgetting that absolute value equations like |x| = 5 have two solutions: x = 5 and x = -5

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|. For any real number, absolute value strips away the sign and returns only the magnitude.

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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