Absolute Value and the Distributive Property

Date: 12/11/2001 at 20:59:11
From: Ira Campbell
Subject: Absolute Value

Is the distributive property applicable to absolute value?

Date: 12/12/2001 at 10:31:49
From: Doctor Peterson
Subject: Re: Absolute Value

Yes and no. You can't blindly apply it as if the absolute value symbol 
were just a pair of parentheses; but you can prove a sort of 
distributive property using the properties of the absolute value.

Remember that

    |a| * |b| = |ab|

Now let's try to manipulate |a| * |b + c|:

    |a| * |b + c| = |a(b + c)| = |ab + ac|

This is the absolute value equivalent of the distributive property. If 
a is non-negative, you can drop the absolute value around it, and you 
get

    a * |b + c| = |ab + ac|   for a >= 0

But don't forget the condition on a. If a is negative, the left side 
of this is negative, and the right side is positive, which won't work 
well.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/