Proof of Absolute Value, Division

Date: 7/16/96 at 18:42:40
From: Alfred T Chu
Subject: Proof of Abs. Value, Division

Hello, I have this problem:

Can you prove...
If y does not equal to 0, then
   |x / y| = |x|/|y|

I can't type it on the computer so I am going to say it.
Prove that:  Absolute value of x over y equals absolute value of x 
over absolute value of y.

Please help me out. Thanks a whole lot!

Alfred

Date: 7/23/96 at 15:25:36
From: Doctor Erich
Subject: Re: Proof of Abs. Value, Division

Alfred,

I not completely sure of your exact question but I'm guessing you want 
to prove that |x/y| = |x|/|y| as long as y doesn't equal zero, 
correct?  

Assuming this is the case there are several helpful things to remember 
when dealing with absolute values. First, sometimes absolute value 
problems can be greatly simplified by dealing with them in two parts -
in this case seeing what happens for y<0 and then seeing what happens 
for y>0.  

In this case, if you could prove the theorem for both these simpler 
cases, you could conclude the theorem held for all y not equal to 
zero.  Another helpful thing to remember when dealing with absolute 
values is that |x|*|y|=|x*y|. For example 

  |2|*|-3|= 6 = |-6| = |2 *(-3)|

You can even do this with powers of x and y.  Unfortunately, these 
manipulations don't work as nicely when you're adding or subtracting 
absolute values.  Anyway, I hope that helps you out some... and I hope 
I guessed the right problem.  Write back if you have any other 
questions.

-Doctor Erich,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/