Absolute Value Equation
An absolute value equation contains an absolute value expression and is solved by considering both positive and negative cases of the expression inside the absolute value.
Formula
|x| = a \text{ means } x = a \text{ or } x = -a
Definition
An absolute value equation has absolute value bars around part of the expression; since absolute value measures distance from zero, the answer could be a positive or a negative number. To solve $|ax + b| = c$ where $c > 0$, set the inside equal to $c$ and to $-c$ and solve both equations; if $c = 0$ there is one solution, and if $c < 0$ there is no solution, since absolute value can never be negative. Formally, the absolute value function $|x| = \sqrt{x^2}$ is a piecewise linear function, $x$ if $x \ge 0$ and $-x$ if $x < 0$, and it is the standard metric on $\mathbb{R}$: $d(a, b) = |a - b|$. An absolute value equation $|f(x)| = g(x)$ is equivalent to $f(x) = g(x)$ or $f(x) = -g(x)$, subject to $g(x) \ge 0$, and in normed vector spaces the absolute value generalizes to the norm.
Example
$|x| = 5$ means $x$ is $5$ units from zero, so $x = 5$ or $x = -5$, since both are $5$ steps away on the number line. Solve $|2x - 3| = 7$: Case 1, $2x - 3 = 7$ gives $x = 5$; Case 2, $2x - 3 = -7$ gives $x = -2$. $|x - 3| < 2$ translates to $-2 < x - 3 < 2$, i.e., $1 < x < 5$, the open ball of radius $2$ centered at $3$ in the metric space $(\mathbb{R}, |\cdot|)$.
Key Insight
Absolute value gives distance, and distance is never negative, so any equation with $|\text{expression}| = a$ (where $a > 0$) has two solutions; always check both solutions by substituting back, and check whether the equation has no solution (negative right side) before splitting into cases. The triangle inequality $|a + b| \le |a| + |b|$ is the defining property of a norm, and every metric space and normed space generalizes this property, making absolute value equations the one-dimensional prototype of a vast theory, foundational to epsilon-delta proofs in real analysis.