Unformatted text preview: rem 2.1. Properties of Absolute Value: Let a, b, and x be real numbers and let n be
an integer.a Then
• Product Rule: |ab| = |a||b|
• Power Rule: |an | = |a|n whenever an is deﬁned
• Quotient Rule: a
, provided b = 0
|b| • |x| = 0 if and only if x = 0.
• For c > 0, |x| = c if and only if x = c or x = −c.
• For c < 0, |x| = c has no solution.
a Recall that this means n = 0, ±1, ±2, . . . . The proof of the Product and Quotient Rules in Theorem 2.1 boils down to checking four cases:
when both a and b are positive; when they are both negative; when one is positive and the other 128 Linear and Quadratic Functions is negative; when one or both are zero. For example, suppose we wish to show |ab| = |a||b|. We
need to show this equation is true for all real numbers a and b. If a and b are both positive, then
so is ab. Hence, |a| = a, |b| = b, and |ab| = ab. Hence, the equation |ab| = |a||b| is the same as
ab = ab which is true. If both a and b are negative, then ab is positi...
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