> [!summary] The note is a shortcut for the absolute values rules and serves as a hub for proofs of rules > $ \begin{array}{|c|l|} \hline \textbf{Rule} & \textbf{Statement} \\ \hline \text{Zero Property} & |x| = 0 \iff x = 0 \\ \hline \text{Multiplication Rule} & |ab| = |a||b| \\ \hline \text{Division Rule} & \left| \frac{a}{b} \right| = \frac{|a|}{|b|},\quad b \ne 0 \\ \hline \text{Power Rule} & |x^n| = |x|^n \quad \text{(if } n \text{ even)} \\ \hline \text{Identity with Square Root} & |x| = \sqrt{x^2} \\ \hline \text{Triangle Inequality} & |a + b| \le |a| + |b| \\ \hline \hline \text{Inequalities Rule} & \begin{array}{c} \text{If $|x| \leq A$ then:} \\ -A \leq x \leq A \\ \\ \text{If $|x| \geq A$ then:} \\ |x| \leq -A \space \lor |x| \geq A \end{array}\\ \hline \end{array} > $ >[!info]+ Read Time **⏱ 1 min** ## Product Rule $ |a||b| = |ab| $ > [!note]- Proof > ![[Proof of the Absolute Value Product Rule]] ## Quotient Rule $ \frac{|a|}{|b|} = \left| \frac{a}{b} \right| \quad (b \neq 0) $ > [!note]- Proof ![[Proof of the Absolute Value Quotient Rule]] ## Power Rule Note that this rule only works for $nth$ power as an [[Integers|integer]]. $ |a^n| = |a|^n \quad n \in \mathbb{Z^+} $ > [!note]- Proof ![[Proof of the Absolute Value Power Rule]] ## Identity with Square Root $ \sqrt{ x^2 } = |x| $ > [!note]- Proof ![[Proof of the Absolute Identity with the Square Root Rule]] ## Triangle Inequality Rule $ |a+b| \leq |a|+|b| $ > [!note]- Proof ![[Proof of the Absolute Value Triangle Inequality]] ## Inequalities Rule $ \begin{array}{c} \text{If $|x| \leq A$ then:} \\ -A \leq x \leq A \\ \\ \text{If $|x| \geq A$ then:} \\ |x| \leq -A \space \lor |x| \geq A \end{array} $ > [!note]- Proof ![[Proof of the Absolute Inequalities Rule]]