Simplify each complex fraction. [ (-2)/(x+h) - (-2)/x ] / h

Question

Accepted Answer

Hello. Everyone in this video we're going to be looking at this practice problem where we want to simplify the expression negative eight divided by a plus c minus negative two divided by A. And all of that is being divided by C. So you can see that this expression has fractions within fractions. So in order to make it easier to simplify, I'm going to start by just looking at the numerator. So the numerator of the big fraction is negative eight divided by a plus c minus negative two divided by A. And in this case you can see that there's two fractions and they have different denominators. So in order to combine them I need to make them have the same denominator and I can do that by multiplying by the other's denominator. So in this case this first fraction has a denominator of A plus C. So I can multiply by the second fractions denominator which is just A. And I want to multiply by a factor of one or that simplifies to one because I don't want to reintroduce any new numbers into my expression. So you can see that a divided by a. It's the same thing as one. And I want to do that to my second fraction as well. Except that instead of multiplying by a over A. I need the first fractions denominator which is a plus C. So multiplied by a plus C. Time divided by a plus C. So once I do this multiplication I have eight times negative eight. That is negative eight A divided by a times a plus c minus negative two times A plus C divided by a times A plus C. And now you can see these two fractions have the same denominators so I can go ahead and combine them. So when I combine them I have negative eight A minus negative two times A plus C. Divided by that common denominator of a times A plus C. And let's go ahead and simplify this numerator a bit. So I have to distribute this negative two. So negative two times A is negative two A negative two times C. Is negative two C. And that's still being divided by a times A plus C. And actually I can also distribute this negative so two negatives make a positive. So this becomes plus two A. We can't forget to distribute it to this back negative two C becomes positive to see. And now we can combine like terms and it seems like we can combine negative A. With positive to A. So that leaves us with negative six A. Plus to see divided by a times A plus C. And you may think that we're done. But recall that all of this is over. See and because I have fractions over fractions or I still have a fraction in the numerator if I write this in the more linear format which is divide by see over one. I can change this into multiplication if I switch the numerator and denominator of my second fraction. So instead of being C over one I'll have one oversee. And with this multiplication I'll have negative six A plus two C. Divided by A. Times C. Times A. Plus C. So you can see I just multiplied the new C into my denominator and it seems like we can't simplify this any further but we can pull out a negative two from our numerator. So if I pull out negative two from negative six AM left with positive three A. If I pull out negative to from positive to see I'm left with negative C. And my denominator stays the same. I have A C. Times A. Plus C. And from here there's no more pulling out that we could do or simplifying. So this becomes our final simplification for the expression and you can see that answer choice A. Also has negative two times three A minus C. Divided by A C. Times A plus C. So hopefully this video helped and see you next time

Simplify each complex fraction. [ (-2)/(x+h) - (-2)/x ] / h

Key Concepts

Complex Fractions

Common Denominator

Algebraic Manipulation

Watch next