Absolute Value Formula

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

The Formula

∣x∣=distance from 0|x|=\text{distance from }0

When to use: −5-5 and 55 are both 5 units from zero, so ∣−5∣=∣5∣=5|-5| = |5| = 5.

Quick Example

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

Notation

∣x∣|x| is never negative because distance is never negative.

What This Formula Means

The distance of a number from zero on the number line, always non-negative; written ∣x∣|x|. For any real number, absolute value strips away the sign and returns only the magnitude.

−5-5 and 55 are both 5 units from zero, so ∣−5∣=∣5∣=5|-5| = |5| = 5.

Formal View

∣x∣={xx≥0−xx<0,equivalently ∣x∣=x2|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∣|{-7}| + |3|.

Answer

1010

First step

1
The absolute value of −7-7 is the distance from 0: ∣−7∣=7|{-7}| = 7.

Full solution

  1. 2
    The absolute value of 33 is ∣3∣=3|3| = 3.
  2. 3
    Add: 7+3=107 + 3 = 10.
Absolute value measures distance from zero on the number line and is always non-negative. For any number aa, ∣a∣=a|a| = a if a≥0a \ge 0 and ∣a∣=−a|a| = -a if a<0a < 0.

Example 2

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

Example 3

medium
Compute ∣∣−3∣−∣−8∣∣\big||{-3}|-|{-8}|\big|.

Common Mistakes

  • Keeping the negative sign inside absolute value — distance from zero is nonnegative.
  • Thinking ∣a−b∣|a-b| equals a−ba-b always — order matters before absolute value removes sign.
  • Using absolute value when direction matters — a temperature of -7 degrees is not the same situation as +7 degrees.

Why This Formula Matters

Absolute value helps students separate direction from size. It supports integer operations, coordinate distance, error bounds, and absolute value equations later. Recognizing it by "Is the sign direction important, or only the distance?" — rather than by familiar numbers — is what lets a student tell it apart from negative numbers and opposites in a mixed problem set.

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

How do you use the Absolute Value formula?

−5-5 and 55 are both 5 units from zero, so ∣−5∣=∣5∣=5|-5| = |5| = 5.

What do the symbols mean in the Absolute Value formula?

∣x∣|x| is never negative because distance is never negative.

Why is the Absolute Value formula important in Math?

Absolute value helps students separate direction from size. It supports integer operations, coordinate distance, error bounds, and absolute value equations later. Recognizing it by "Is the sign direction important, or only the distance?" — rather than by familiar numbers — is what lets a student tell it apart from negative numbers and opposites in a mixed problem set.

What do students get wrong about Absolute Value?

The procedure for absolute value is the easy part; the trap is keeping the negative sign inside absolute value. Asking "Is the sign direction important, or only the distance?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Absolute Value formula?

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

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