Modulus Equations & Functions

At a glance 11 formulas

1. 1Definition$|x| = x \ (x \ge 0),\quad |x| = -x \ (x < 0)$$\sqrt{x^2} = |x|,\quad |x|^2 = x^2$
2. 2Distance$|x - a| = \text{distance from } x \text{ to } a$$|a - b| = |b - a|$
3. 3Modulus equation$|x - a| = b \Rightarrow x = a \pm b \ (b \ge 0)$$b < 0 \Rightarrow \text{no solution}$
4. 4Less than, more than$|x - a| < b \iff a - b < x < a + b$$|x - a| > b \iff x < a - b \text{ or } x > a + b$
5. 5Bars on both sides$|A| = |B| \iff A = \pm B$$|A| < |B| \iff A^2 < B^2 \ (\text{any } A, B)$
6. 6Critical pointsMark where each bar's inside is 0, split the number line there, and drop the bars piece by piece. Keep only answers that lie in their own piece.
7. 7Sum of two distances$|x - a| + |x - b| \ge |a - b|$$\text{equal for every } x \text{ between } a \text{ and } b$
8. 8Median rule$|x - a_1| + |x - a_2| + \dots + |x - a_n|$ is least at the median of the $a$'s; with an even count, anywhere between the middle two.
9. 9Triangle inequality$|a + b| \le |a| + |b|$$\text{equal exactly when } ab \ge 0$
10. 10Graph of |f(x)|Draw $y = f(x)$, then flip every part below the $x$-axis up. The solutions of $|f(x)| = k$ are where the line $y = k$ cuts this graph.
11. 11Diamond area$|x - p| + |y - q| = c:\ \text{area } 2c^2$$a|x| + b|y| = c:\ \text{area } \dfrac{2c^2}{ab} \ (a, b, c > 0)$