Simplify each complex fraction. See Examples 5 and 6.

y + 3 4
...

Question

Simplify each complex fraction. See Examples 5 and 6.

y + 3 4
 ———— - ————
 y y - 1
——————————
 y 1
 ——— + ——
 y - 1 y

Accepted Answer

Hello, everyone. We are asked to simplify the following rational expression. We are given X plus two divided by X minus five, divided by X minus three. All of which is divided by X divided by X minus three plus one, divided by X. Our answer choices are A X or in parentheses, X squared plus three divided by the parentheses, X minus three, B in parentheses, X squared minus six, X minus six divided by the parentheses. X squared plus X minus three C, the parentheses, X squared plus six, X minus six divided by the parentheses, X squared plus three, X minus three, DX squared plus three, X plus six. So rewriting this again, I have X plus two divided by X minus five, divided by X minus three, all of which is divided by X divided by X minus three plus one divided by X. So I do need to make a common denominator and there are a few different ways we can do this. But I'm gonna do the numerator and the denominator separately initially. So for my numerator, my common denominator would be X multiplied by the parentheses X minus three. So I'm gonna set up two fractions with that as my denominator in the numerator of my bigger fraction and to go from X to X multiplied by X minus three, I must multiply it by X minus three in parentheses. So I also have to multiply the numerator by X minus three, which will give me X plus two in parentheses, multiplied by X minus three in parentheses. I'm gonna change my middle minus sign to a plus and make the five in the numerator that follows it into a negative five. And now I'm gonna ask myself, what did I multiply the denominator by to get X multiplied by X minus three. And the answer is X. So I'm gonna multiply numerator and denominator of my second fraction by X. So at this step, I now have the parentheses, X plus two multiplied by the parentheses, X minus three divided by X multiplied by X minus three plus negative five, X divided by X multiplied by X minus three. I'm gonna do the same with the denominator before I expand my numerator. So in the denominator of my bigger fraction, it looks again like I will have a common denominator of X multiplied by the parentheses X minus three. So I'm setting up those two fractions. So the first fraction in the denominator to get from X minus three to X multiplied by X minus three. I have to multiply numerator and denominator by X and X multiplied by X is X squared for the second fraction I need to multiply numerator and denominator by the parentheses X minus three. So one multiplied by X minus three is just X minus three. And now I can expand things and combine like terms. So in order to keep this linear off to the side, I am going to multiply out X plus two multiplied by the parentheses X minus three using foil just to see what it is. And then I'll plug it back in. So first gives me X squared outer negative three, X inner two, X last negative six. So expanded, this is X squared minus X minus six. So I'm gonna put that in my numerator. So X squared minus X minus six divided by X multiplied by X minus three plus negative five, X divided by X multiplied by X minus three in parentheses divided by. And I'm gonna combine into one fraction in my denominator right now. So I'll have X squared plus X minus three divided by X multiplied by X minus three. I need to make my numerator into one fraction. So I'm gonna do that by combining my like terms. And I'll have X squared minus six, X minus six divided by X multiplied by X minus three in parentheses. And that's divided by X squared plus X minus three, divided by X multiplied by X minus three. So here, since I have two fractions that have the same denominator, I can multiply the full numerator fraction and the full denominator fraction by X multiplied by the parentheses X minus three. And when I do these things will cross reduce. So we are left with X squared minus six, X minus six divided by X squared plus X minus three. Now, I wanna ask myself, can this be factored any further? Like do those trinomial break down to binomials? And I think the denominator might, but my numerator I'm not coming up with anything that's gonna multiply to negative six and add to negative six. So I don't think that can be factored any further. So this is my answer X squared minus six, X minus six divided by X squared plus X minus three. And that's the answer choice. B have a nice day.

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Simplify each complex fraction. See Examples 5 and 6. y + 3 4 ———— - ———— y y - 1 —————————— y 1 ——— + —— y - 1 y