Absolute Values

Have you ever described how far away something is? When you say "the supermarket is 3 km away," you never say "negative 3 km" , distance is always positive. That's exactly the idea behind absolute value.

Absolute Value: The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. We write it using vertical bars: $|x|$, read as "the absolute value of $x$".

For example:

• $|5|=5$ (5 is already 5 steps from zero)
• $|-5|=5$ (−5 is also 5 steps from zero, just in the opposite direction)
• $|0|=0$ (zero is zero steps from zero)

Absolute value is defined as a piecewise function , meaning it has two different rules depending on the value of $x$:

$|x|=\{\begin{array}{ll}x& \text{if }x\ge 0\\ & \\ -x& \text{if }x<0\end{array}$

In plain English:

• If $x$ is positive or zero, the absolute value is just $x$ itself.
• If $x$ is negative, the absolute value is $-x$ (which flips the negative to a positive).