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### 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

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

```
Associated Topics:
High School Basic Algebra

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