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