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MATH LIBRARY

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:
         |ab| |a| – |b|

CAREFUL!!    Sums:   

Differences:   

|a + b| is not the same as |a| + |b|

|ab| is not the same as |a| – |b|

 

See also

Absolute value of a complex number, triangle inequality (geometric)

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