Doubling Opposite Integers

Date: 04/19/98 at 13:49:55
From: Renee Bailey
Subject: Integers 

On a number line, a certain integer is 24 units away from the double 
of its opposite.  What is the integer?

I started to look at different integers and the double of their 
opposites, for example 12 and 21(2), which are 30 units apart, and 43 
and 34(2), which are 25 units apart, etc. However, I did this with 
about 50 different numbers and still have not found two that are 24 
units apart.  

So I was wondering, is there an equation that I can use to solve this 
problem, and if so what? Or do I have to keep doing trial and error?

Date: 04/19/98 at 14:03:16
From: Doctor Rothman
Subject: Re: Integers 

Hi Renee.  

Let's actually draw a number line and look at where the integer, its 
opposite, and the double of its opposite might lie. 

Remember that the "opposite" of a number implies that you multiply the 
number by -1, not that you reverse the digits.  Let's call the integer 
P, because we'll start out assuming it's positive (although the 
problem doesn't say it has to be).  So here's the number line:

-2P            -P               0                P
-|-------------|----------------.----------------|-----------------

P is our integer, -P is its opposite, and -2P is the double of its 
opposite.

If we then say that the distance between P and -2P is 24, then
P - (-2P) = 24.

Now solve this equation for P.  When you check this value for P, is it 
indeed 24 units away from "double its opposite," -2P?

There's still another part to this problem though. We assumed the 
integer was positive, but what if it was negative? I'm going to let 
you figure that out on your own.

Good luck!

-Doctors Naomi and Rothman,  The Math Forum
 http://mathforum.org/dr.math/