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### Adding and Subtracting Algebraic Fractions

```
Date: 8/24/96 at 5:38:34
From: D.S.K.

Hi Dr. Math! I am having a little trouble with the subject of adding,
subtracting, and simplifying fractions with the same denominator. I
hope you can help me out.  My problem is that I am not sure of my
method. Could you please take a look and see what you think?

The instructions are to perform the indicated operations and express
the results in lowest terms. Here is the first one:

5       4x      2-x
----  -  ----  -  ----
x-1     x-1      x-1

Here is the way I did it:

5-4x-(2-x)     3-3x
----------  = ------  At this point I am not sure if the problem is
x-1         x-1    finished or if I can  do any more to it. I
thought perhaps that I could proceed like this:

3(1-x)      3
---------  = ---  =  -3.  Is this right?
(-1)(1-x)    -1

Here is another one:

2     2            2          2
r  - 3s     2rs    r - 2rs - 3s    (r-3s)(r+s)     r-3s
-------  -  --- =  -----------  =  ----------  =  -----  =  r-3s.
r+s        r+s          r+s         r+s            1

This one really had me going, because in the second step I had to
rearrange the terms in the numerator in order to get the right answer
and I was not sure if I was allowed to do that. Is this okay to do?
```

```
Date: 8/25/96 at 0:32:37
From: Doctor Mike
Subject: Re: Adding and Subtracting  Fractions

Hello DSK,

First of all, always remember that an expression like 5/(x-1)
only makes sense when the denominator is not zero, which in this
case means that x is different from one.

What you did in both problems is exactly what I would have done.
Any factoring or multiplying out or rearranging that does not
change the meaning is fine.  You seem to have this subject down.

The next stage of problems are ones where the denominators are
not all the same. Then you cannot just add/subtract numerators.
You have to do some work to find a common denominator so that the
fractions can be added as above.  For example, consider:

5         2x
------- - --------
x+1      x(x+1)

You cannot add these as-is because the denominators are different.
You can, however, make the 2 denominators the same by multiplying
the 5/(x+1) fraction by one .... not just any old one, but by x/x
which is of course equal to one.  Then you have

5x         2x
-------- - --------  = .... you figure it out from here.
x(x+1)     x(x+1)

Keep up the good work.

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

```
Date: 8/25/96 at 5:56:15
From: D.S.K.
Subject: Re: Adding and Subtracting Fractions

Thank you for helping me out, Doctor Mike. You sure have saved me from
a lot of brain-racking! And thanks for the info on adding/subtracting
fractions with different denominators, that is what I will be studying
next.

D.S.K.
```

```
Date: 8/26/96 at 10:27:37
From: Doctor Mike
Subject: Re: Adding and Subtracting Fractions

Hello again,
You're very welcome. Let us know if you need another boost.

-Doctor Mike,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Algebra
Middle School Fractions

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