Absolute Value

Definition

The absolute value of a number is its distance from zero on the number line.

Symbol: $|a|$

Key point: Distance is always positive (or zero)!

Examples

• $|5| = 5$ (5 is 5 units from zero)
• $|-5| = 5$ (-5 is also 5 units from zero)
• $|0| = 0$ (0 is 0 units from zero)

Formal Definition

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

Opposite Numbers

Numbers like 5 and -5 are opposites (same absolute value, different signs).

If $|a| = |b|$ and $a \neq b$, then $a = -b$

Absolute Value with Operations

Evaluate inside first, then take absolute value:

• $|3 + (-7)| = |-4| = 4$
• $|-6| + |2| = 6 + 2 = 8$
• $|5 - 8| = |-3| = 3$

Comparing Absolute Values

To compare $|-8|$ and $|5|$: $|-8| = 8, \quad |5| = 5$ $8 > 5$