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Absolute Value Rules

Algebra rules for absolute values are listed below.

 Piecewise Definition: $$\left| a \right| = \left\{ {\begin{array}{*{20}{c}}a&{{\rm{if}}\;a \ge 0}\\{ - a}&{{\rm{if}}\;a < 0}\end{array}} \right.$$ Square root definition: $$\left| a \right| = \sqrt {{a^2}}$$ Rules: 1.  |–a| = |a| 2.  |a| ≥ 0 3.  Products: |ab| = |a||b| 4.  Quotients: |a / b| = |a| / |b| 5.  Powers: |an| = |a|n 6.  Triangle Inequality:          |a + b| ≤ |a| + |b| 7.  Alternate Triangle Inequality:          |a – b| ≥ |a| – |b| CAREFUL!! Sums: |a + b| is not the same as |a| + |b| |a – b| is not the same as |a| – |b|