Simplifying Equations

Date: 4/11/96 at 21:58:53
From: Anonymous
Subject: Algebra

I am having trouble on the following problems. I need to solve 
them by factoring, finding the lowest common denominator, then 
multiplying by the reciprocal.

Simplify:

(1+(1/x-1))/(1+(1/x^2-1))

and expressing f(x+h)-f(x)/h as a single simplified fraction for 
the following exercise:

f(x) =(1-x)/x

Date: 4/14/96 at 0:52:33
From: Doctor Patrick
Subject: Re: Algebra

Hi.  For the first problem, what you need to do is simplify the 
terms within the parentheses first, then go on to work on the rest 
of the problem.

First we look at the 1+(1/x - 1).  1/x - 1 needs to be rewritten 
so that we have the same denominator for both terms.  This is easy 
to do since we can change the 1 to x/x (do you understand why?), 
making our problem 1+ (1/x - x/x).  This is much easer to deal 
with, don't you agree? Now, we simplify 1/x - x/x by subtracting 
the fractions.  When we do this we get (1-x)/x.

Since this is as simple as we are going to be able to make this 
term, we can move on to the rest of the problem.   We now have 
1+ (1-x)/x.  We can simplify this in the same way we did the last 
step - by finding a common denominator.  Again, we change the 
1 to x/x so that we can add fractions.  This time we get 
x/x + (1-x)/x), or (x+1-x)/x.  When we subtract like terms we get 
1/x, a much nicer number to work with!

This makes the problem (1/x)/(1+(1/x^2-1)).  I'll let you simplify 
the (1+(1/x^2-1)) on your own, using the same steps we did before.  
The only difference is that you will be using x^2 instead of x.  
When you simplify this part (it should also come out to be a 
simple number) you will have to divide the first part by the 
second part to get your final answer. 

 Good luck!

-Doctor Patrick,  The Math Forum

Date: 4/14/96 at 1:0:4
From: Doctor Ken
Subject: Re: Algebra

Hello there!

I assume that you're doing your second problem in the context of 
takingderivatives, right?  Or do you not know about these yet?  In 
any case, the expression [f(x+h)-f(x)]/h   is one that comes up 
all the time in calculus.  

The first thing I think I'd do is make things a little less 
confusing for myself.  You've got two x's running around there, 
one in the definition of f(x), and the other in the expression 
you're trying to simplify.  I'm going to write the function f in 
terms of t, not x, like this:

f(t) = (1-t)/t.

Then if we plug in our function to the expression [f(x+h)-f(x)]/h, 
we get

1 - (x+h)      1 - x
---------  -  -------
   x+h           x
-----------------------
           h

Now we can do some simplifying.  The first thing I'd do is work on 
that numerator (the top), and get it into one fraction.  To do 
that, you'll find a common denominator and do the subtraction.  Do 
you think you can take it from here?  

Good luck, and if you need more help on this, you can write back.

-Doctor Ken,  The Math Forum