Absolute Value Equations & Inequalities

Distance from zero: split equations into two cases and inequalities into a band or two rays.

The idea

Definition (Absolute value).

The absolute value $|X|$ of a quantity $X$ is the distance from $X$ to $0$ on the number line. Here $X$ may be any expression, not only a single letter.

A quantity $X$ and its opposite $-X$ lie the same distance out, so $|4| = |-4| = 4$, and $|X|$ is never negative.

Every equation and inequality built with absolute-value bars is a statement about distance, and three rewrites remove the bars.

Proposition.

Let $c$ be a positive number. Then $|X| = c$ exactly when $X = c$ or $X = -c$; $|X| < c$ exactly when $-c < X < c$; and $|X| > c$ exactly when $X < -c$ or $X > c$.

$|X| = c$ says $X$ lies exactly $c$ units from $0$. Two points do, one on each side, so $X = c$ or $X = -c$.

$|X| < c$ says $X$ lies less than $c$ units from $0$. Those points fill the interval between $-c$ and $c$, so $-c < X < c$.

$|X| > c$ says $X$ lies more than $c$ units from $0$: everything to the left of $-c$ together with everything to the right of $c$, so $X < -c$ or $X > c$.

Rewriting the statement in one of these forms removes the bars, and ordinary solving finishes the problem.

Ways to work on it

• Walkthrough. Read |X| as distance, split equations into two cases, and tell < (a band) from > (two rays).
• Practice. Solve a randomized absolute value equation.
• Hardest. Solve an absolute value inequality and handle the no-solution edge cases.

Not sure where to start? Take the ten-question placement test.