simplifying fractions...?

I am going to assume that the problem reads as such
3/(x-1) - (4x)/(1-x)

When adding/subtracting fractions, the most important thing to remember is that they must have a common (same) denominator. (x-1) and (1-x) are not the same, so we will change them!

Let's use a different example for a minute. If we had 1/3 - 1/5, to find a common denominator we would multiply the 3 and 5 together, right?

Same thing in this case. Your common denominator will be (x-1)(1-x). Don't worry about actually multiplying them together, you can write them just like that.

Now... we have found our common denominator but need to change the original fractions, right? When we look at the first one, the denominator already has an (x-1), so we know we must multiply the top and bottom by (1-x). The second term already has a (1-x) in the denominator so we know we must multiply the top and bottom by (x-1). When we do this, our answer reads

3(1-x) - (4x)(x-1) all divided by (x-1)(1-x).

We aren't quite done, though. I have a sneaking suspicion that if we distribute the top out, there will be like terms we can combine... let's see...

The top, after distributing, will read 3 - 3x - 4x^2 + 4x. Don't forget to distribute the negative with that 4x! We can combine the -3x and +4x to make plain x. Our final answer is (drum roll, please...)

4x^2 + x + 3 all divided by (x-1)(1-x)

Taadaa!!!

The answer provided by stephiquinone (the algebra teacher) is completely correct except for one thing....After reminding you to distribute the negative, she/he forgot to include the negative in their final answer.

In this earlier step, as provided by stephiquinone, we must distribute the negative in the middle (i.e. the negative that you see between 3(1-x) and (4x)(x-1):
3(1-x) - (4x)(x-1) all divided by (x-1)(1-x)

Distributing this negative yields a -4x^2 (because -4x times x = -4x^2) and a +4x (because -4x times -1 = +4x).

All of this work done by stephiquinone was correct, but then she/he mistakenly said that the final answer should be:
4x^2 + x + 3 all divided by (x-1)(1-x)

But, in actuality, the final answer should be:
-4x^2 + x + 3 all divided by (x-1)(1-x)

We caught the teacher! :)

3/x - 1 - 4x/1 - x
4x = xxxx
xxxx/ x = cross out x on the bottom and cross out 1 x from the top so it becomes
xxx= 3x
3x- x=2x
So, we have:
3/2x -1 /1
3-1 = 2. 2/ 1= 1.
So we now have 1/2x! Thus the Answer!
I hope this helped! Good luck learning! They are easy when you get the hang of them!

....3..........4x
---------- - ------------
...x-1.........1-x

....3..........4x
=-------- - ------------
...x-1....-..(x-1)

....3..........4x
--------- + ----------
..x-1..........x-1

......4x+3
=------------- answer//
.......x-1

3/(x - 1) - (4x)/(1 - x)
= 3/(x - 1) - (4x)/[-(-1 + x)]
= 3/(x - 1) - [-(4x)]/(-1 + x)
= 3/(x - 1) + (4x)/(x - 1)
= (3 + 4x)/(x - 1)
= (4x + 3)/(x - 1)