Simplify each complex rational expression. [3-1/(x+3)]/[3+1/(x+3)...

Question

Simplify each complex rational expression. [3-1/(x+3)]/[3+1/(x+3)]

Accepted Answer

Hello today we're gonna be working on simplifying irrational expression. So the rational expression given to us is two plus one over X plus two. All over two minus one over X plus two. So before we start simplifying the rational expression let's go ahead and simplify our numerator and denominator separately. So in the numerator we have two plus one over X plus two. Now we can go ahead and imagine to to be over one to make it a fraction. And in order to combine these together we need to go ahead and get the lowest common denominator. Well we have the quantity X plus two in this fraction and just one in the other and one can be anything you want. So let's go ahead and make the L. C D. X plus two. So in order to get the lowest common denominator of X plus two in the left hand side we're gonna multiply the numerator and denominator by X plus two. Doing so is going to give me two times X plus two over X plus two plus one over X plus two. Now both of these fractions have the same denominator so let's go ahead and combine them into a single fraction. Doing so is going to give me two times X plus two plus one over X plus two. Now in order to simplify this numerator I'm going to distribute the two into X-plus two. Doing so is gonna give me two X plus four plus one over X plus two and four plus one is going to give me two. X plus five over X plus two. So that's gonna be our numerator simplified next let's go ahead and simplify our denominator. Our denominator is two minus one over X plus two. Now just like we did with the numerator, we're gonna rewrite two as a fraction by putting it over one And the lowest common denominator between these two fractions is going to be X-plus two. So in order to get X plus two on the left hand side we're gonna multiply the numerator and denominator by X plus two. Doing so is going to give me two times X plus two over X plus two minus one over X plus two. And since both of these fractions have the same denominator we can go ahead and combine them into a single fraction. Doing so is gonna give me two times X plus two minus one. All over X plus two. Now to simplify the numerator, I want to distribute this two into X plus two. Doing so is going to give me two, X plus four minus one over X plus two and four minus one is going to give me three, leaving me with two, X plus three over X plus two. So now we have a simplified numerator and a simplified denominator. So let's go ahead and put this back into the complex fraction. The numerator is two, X plus five over X plus two. And all of that is going to be over two X plus three Over X-plus two. Now we can simplify the complex fraction by multiplying by the reciprocal of the denominator. The reciprocal of the denominator is going to be X plus 2/2 X plus three. And we're gonna multiply the reciprocal to the numerator as well. So that's gonna be X plus 2/2, X plus three. The reason why we do this is because this allows us to completely cancel the numerator and denominator down to just one and that essentially cancels out the complex fraction. That then leaves us with two X plus five times X plus two over X plus two times two X plus three. Now, if you notice we have X plus two in the numerator and denominator so we can go ahead and cancel out those quantities. That then leaves us with two X plus 5/2 X plus three. And since this is the simplest form of this complex fraction, this is going to be our solution. This also means that the solution to our problem is going to be be so I hope this video helps you simplify complex fractions and I'll go ahead and see you all in the next video

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