Solve each equation in Exercises 83–108 by the method of your cho...

Question

Solve each equation in Exercises 83–108 by the method of your choice.

3/(x - 3) + 5/(x - 4) = (x^2 - 20)/(x^2 - 7x + 12)

Accepted Answer

Hello. Today we're going to solve the given equation using any of our preferred methods. So the equation given to us is X over the quantity X minus four plus four. Over the quantity X plus four is equal to negative 23 over X squared minus 16. Now before we start this we're gonna go ahead and simplify the right hand side. Specifically the denominator because X squared minus 16 is a difference of squares and that's going to factor two X minus four times X plus four. So rewriting this equation is going to give us X over x minus four plus four over X plus four is equal to negative 23 over X plus four times x minus four. Now, what we want to go ahead and do is eliminate all of the denominators. And in order to do that we're going to multiply every term here by the lowest common denominator. While the lowest common denominator is going to be the product of all the denominators combined. And the lowest common denominator for this equation is going to be X plus four times X minus four. So with that being said we're gonna multiply the entire equation by X plus four Times X -4 and doing so is going to give us X times X plus four Plus four times X -4 is equal to -23. Now what I'm gonna go ahead and do is open up the parentheses on the left hand side of this equation. And in order to do that I'm going to distribute X two X plus four and distribute +42 x minus four. Doing so is going to give me X squared plus four X plus four X minus 16 is equal to negative 23 and I'm gonna go ahead and combine like terms here well four X and four X can be combined together and that's going to give me X squared plus eight, X minus 16 is equal to negative 23. And since we are solving for X We're gonna go ahead and set this equation equal to zero. And in order to do that we're gonna go ahead and add 23 to both sides of the equation. Doing so is going to give me X squared plus eight, X plus seven is equal to zero. Now this is in the form of a fact herbal trin Amiel because this try no meal is going to factor two, X plus one times X plus seven and that is going to be equal to zero. Now we're gonna use our zero property to set both of these quantities equal to zero so what we get is X plus one is equal to zero and X plus seven is equal to zero and in order to solve for both quantities of X, I'm gonna subtract one to both sides of this equation and seven to both sides of this equation and what I am left with is X is equal to negative one and X is equal to negative seven and these are gonna be the solutions to this equation. And since X is equal to negative one and negative seven, the solution to this problem is going to be C. So I hope this video helps you understanding how to solve this type of equation, and I'll go ahead and see you all in the next video.

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