Solve each equation. See Example 2. x/(x-1) - 1/(x+1) = 2/(x^2-1)

Question

Accepted Answer

Hey everyone welcome back in this problem for the following equation we're asked to solve for X. Were given X over x minus two minus two over X plus two is equal to eight over x squared minus four. So rewriting what we're given X over x minus two minus two. Over X plus two is equal to eight over x squared minus four. Now, when we have in equation like this and we're asked to solve for X, what we wanna do is we want to try to combine these terms were given okay, we want to get them into one term or one rational expression. Okay, so that we can compare it maybe to zero having zero on one side. One equation on the other side makes it a little bit easier to solve for X. Okay, so in order to do that, what we need to do is find a common denominator. Okay, just like if you were adding and subtracting fractions with numbers, we have these fractions with various functions. We need a common denominator so that we can combine them. So Let's see we have a denominator of X -2 on the far left term, the second term on the left hand side, we have a denominator of X plus two. And then on the right hand side we have a denominator of x squared minus four. Now notice that X squared minus four, this is a difference of squares so we can write this as X plus two times X minus two. And so we see that our common denominator will be just that X plus two times X minus two. So starting from the left we have X. Now in this term the denominator x minus two. We want to multiply this by X plus two so that we have that common denominator. So we multiply the numerator by X plus two. Then in the denominator we can also multiply by X plus two. Okay because we have X plus two divided by X plus two, that's just one. So we're just multiplying by one which doesn't change our equation. Alright then the next turn okay we have two over X plus two. So we have X plus two in the denominator. We also want to multiply by x minus two. So we multiply the numerator By X -2 and the denominator by X -2. So again we're multiplying just by one, which doesn't change our equation. Alright and then on the right hand side we have eight over X squared minus four. We leave this alone because it already has that denominator now that we have a common denominator, we can combine all of these terms together into one fraction. So let's move everything to the left hand side and do just that. So we have X times X plus two. That's going to be X squared plus two X. And we're subtracting two times X minus two. So we're gonna have minus two X. And then we have minus negative four. So that's gonna be plus four. And then we're gonna subtract eight, bringing this term on the right hand side to the left hand side and all of this is going to be divided by our common denominator, X minus two times X plus two. And it's all equal to zero. Alright, so let's simplify that numerator and see where that takes us. So we have the X square term. We have two x minus two X. So those are going to cancel out you're gonna add to zero. Then we have 4 -8. So we get -4 over X -2 times x plus two Which is equal to zero. Now remember X squared minus four. That's that original denominator that we broke into x minus two. X plus two. So we can actually write X squared minus four. X minus two times X plus two. Okay, so that we have x minus two time X plus two divided by x minus two times X plus two. All equal to zero. Well this fraction on the left hand side is actually equal to one. Okay, It's something divided by itself. We know that's equal to one. So we end up with one is equal to zero. Well, that doesn't look good. Okay, this is not a true statement. And let me change that from not a church statement to saying this is never a true statement. Okay, It doesn't matter what X value you choose. This will never one will never be equal to zero. So this is false. And what this tells us is that we have no solutions. Okay so our original equation has no solutions because we end up with this false statement. Now I want to point something out Okay because you may have approached this differently Okay from the stage here where we have X squared plus two X minus two. X plus four minus eight over x minus two, X plus two is equal to zero or from the step just below it, X squared minus four over that denominator is equal to zero. You may have multiplied that denominator up to the right hand side. Okay to be zero then you're gonna be left with just the numerator and you're going to solve For those routes and you're going to get answers. Okay? So you might be thinking well if I do this a different way, I'm going to get answers. You will but if you do that, the answers you get are going to be positive and negative two. Okay? You always have to check your restrictions. Okay? If you get positive and negative two, if you plug that into your original equation, you're going to have a divide by zero. Okay, so we can't do that either. So if you decided to multiply up from that step, you would find potential solutions but when you check your restrictions, you would find that they aren't solutions either. Okay, so either way you do this, you're getting that same answer that we have no solutions. So we go back up to our answer choices. Okay? We see that for this particular equation, there is no solution we have answered. D. Thanks everyone for watching. I hope this video helped see you in the next one.

Solve each equation. See Example 2. x/(x-1) - 1/(x+1) = 2/(x^2-1)

Key Concepts

Rational Expressions

Common Denominator

Factoring Polynomials

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