Absolute Value Equation
Pre-AlgebraAn 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. Absolute value measures distance from zero, so the answer could be a positive or a negative number.
Example
$|x| = 5$ means $x$ is $5$ units from zero. So $x = 5$ or $x = -5$, because both are $5$ steps away from zero on the number line.
Key Insight
Absolute value gives distance, and distance is never negative. Any equation with $|\text{expression}| = a$ (where $a > 0$) has two solutions because two points are that distance from zero.
Definition
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. If $c < 0$, there is no solution (absolute value can never be negative).
Example
Solve $|2x - 3| = 7$. Case 1: $2x - 3 = 7$, so $2x = 10$, $x = 5$. Case 2: $2x - 3 = -7$, so $2x = -4$, $x = -2$. Solutions: $x = 5$ or $x = -2$.
Key Insight
Always check both solutions by substituting back. Also check whether the absolute value equation has no solution (negative right side) before splitting into cases.
Definition
The absolute value function $|x| = \sqrt{x^2}$ is defined as a piecewise linear function: $x$ if $x \ge 0$, $-x$ if $x < 0$. 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 $g(x) \ge 0$. In normed vector spaces, the absolute value generalizes to the norm.
Example
$|x - 3| < 2$ translates to $-2 < x - 3 < 2$, i.e., $1 < x < 5$. This is the open ball of radius $2$ centered at $3$ in the metric space $(\mathbb{R}, |\cdot|)$. Absolute value equations and inequalities are foundational in epsilon-delta proofs in real analysis.
Key Insight
The triangle inequality $|a + b| \le |a| + |b|$ is the defining property of a norm. Every metric space and normed space generalizes this property, making absolute value equations the one-dimensional prototype of a vast theory.