Absolute Value

$|x|$ means “the distance from $x$ to 0 on the number line,” which is always nonnegative. That single idea — distance — resolves almost every absolute value problem you'll see, if you translate the symbols into a picture before you start solving.

The piecewise definition

More generally, $|x - a|$ is the distance between $x$ and $a$. That reading makes equations and inequalities involving absolute value much easier to picture:

• $|x - a| = r$ means $x$ is exactly $r$ away from $a$: so $x = a - r$ or $x = a + r$.
• $|x - a| < r$ means $x$ is within $r$ of $a$: the interval $(a-r,\ a+r)$.
• $|x - a| > r$ means $x$ is farther than $r$ from $a$: two rays, $x < a-r$ or $x > a+r$.

When the inside isn't simple

Once the expression inside the bars is more complicated than a single variable, the distance picture stops being immediate, and you fall back on the case split: consider the case where the inside is$\geq 0$ (bars come off unchanged), then the case where it's negative (bars come off with a sign flip). Solve each case as its own equation, restricted to the region where that case is actually valid — and always plug candidate solutions back into the original equation, since squaring or case-splitting can introduce extraneous roots.

Practice problems