Absolute Value - Algebra Cheat Sheet

Definition

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

$|a| = a \quad \text{if } a \geq 0$

$|a| = -a \quad \text{if } a < 0$

Property	Formula
Non-negativity	$\lvert a \rvert \geq 0$
Positive definiteness	$\lvert a \rvert = 0$ if and only if $a = 0$
Symmetry	$\lvert -a \rvert = \lvert a \rvert$
Multiplicative	$\lvert ab \rvert = \lvert a \rvert \cdot \lvert b \rvert$
Division	$\left\lvert \frac{a}{b} \right\rvert = \frac{\lvert a \rvert}{\lvert b \rvert}$ , $b \neq 0$
Triangle Inequality	$\lvert a + b \rvert \leq \lvert a \rvert + \lvert b \rvert$
Reverse Triangle	$\lvert a - b \rvert \geq \lvert \lvert a \rvert - \lvert b \rvert \rvert$
Square Root	$\lvert a \rvert = \sqrt{a^2}$

$|x| = a \quad \text{(where } a \geq 0\text{)}$

Solution: $x = a$ or $x = -a$

$|f(x)| = a \quad \text{(where } a \geq 0\text{)}$

Solution: $f(x) = a$ or $f(x) = -a$

$|f(x)| = |g(x)|$

Solution: $f(x) = g(x)$ or $f(x) = -g(x)$

$|x| < a \quad \text{(where } a > 0\text{)}$

Solution: $-a < x < a$

$|x| \leq a \quad \text{(where } a \geq 0\text{)}$

Solution: $-a \leq x \leq a$

$|x| > a \quad \text{(where } a \geq 0\text{)}$

Solution: $x < -a$ or $x > a$

$|x| \geq a \quad \text{(where } a \geq 0\text{)}$

Solution: $x \leq -a$ or $x \geq a$

The expression $|a - b|$ represents the distance between $a$ and $b$ on the number line.

$|x - c| < r \quad \text{means } x \text{ is within distance } r \text{ of } c$

Solution: $c - r < x < c + r$

$|x - c| > r \quad \text{means } x \text{ is more than distance } r \text{ from } c$

Solution: $x < c - r$ or $x > c + r$

The graph of $y = |x|$ is a V-shape with:

$y = |x - h| + k$

$y = -|x - h| + k$

$y = a|x - h| + k$