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Modulus Cheat Sheet

Keywords: absolute value

Inequalities

Property 1. Let

\[\abs{a - b} \lt \abs{c}\]

then

\[\abs{b} - \abs{c} \lt \abs{a} \lt \abs{b} + \abs{c}\]

Property 2. Suppose that for every $0 \lt \epsilon \lt 1$, the following holds:

\[\frac{b}{1 + \epsilon} \lt \abs{a} \lt \frac{b}{1 - \epsilon}\]

then, for every $\delta \gt 0$:

\[\abs{a - b} \lt \delta\]