In Exercises 61–66, find all values of x satisfying the given con...

Question

In Exercises 61–66, find all values of x satisfying the given conditions.

y1 = (2x - 1)/(x^2 + 2x - 8), y2 = 2/(x + 4), y3 = 1/(x - 2), and y1 + y2 = y3.

Accepted Answer

So this problem tells us to find C. Such that it satisfies the following conditions. We have our conditions there. The main part of this is looking at our final condition. Y one plus y two equals Y three, so Y one plus Y two equals Y three. If you notice we actually have every single one of those variables. Why one is our first equation one over C -1. Y two is our second equation to oversee plus two. And finally why three is our final equation five, C plus six over C squared plus c minus two. So you want to solve for C to do this, we're gonna have to find the least common denominator. But before we do that we can actually do a little bit of factory to make this a little bit easier. You notice here you have a c squared Plus C -2 ashes for this up. And factors so we have a c scripts, we're gonna have a C times something and a C times something. We also have a two at the end, so we're gonna have a one and a two somewhere. So you figure out which sign is gonna give us which, so I'm gonna say we have a plus two and minus one based on what we have in our fractions over here on the left. So if we check it real quick, we would also get c squared C Times to see neither one time C is minus C and they the one times two is negative two, which is the c squared plus c minus two. So those are the factors for R c squared plus c minus two. I can now rewrite this equation one over c minus one plus two over C plus two that is equal to five C plus six. Over our factors c minus one C plus two. Now I want to use the L C. D. To get rid of the fractions everywhere. So we can say first R L C. D is going to be C plus one plus one, c minus one Time C Plus two. So we're gonna multiply everything by C -1, c plus two. If you look here you'll find this c minus one C Plus two on the set. On the other side. Be a multiply by sea minus one C Plus two. And also here C -1 C Plus two. So we're gonna notice that some things are going to cancel out based on the fractions, feel like here have a c minus one on top c minus one on the bottom. Those will cancel same thing here, C plus two in the bottom C plus two on the top. And finally on the right side C minus one times C plus two on the top and c minus one times C plus two on the button. So let's go ahead and rewrite this again. We now have c plus two times one plus two. Time C minus one equals five, C plus six. We can go ahead and distribute and multiply. We have c plus two here plus two times c minus one. We get to see minus two equals five. C plus six. Now we can combine all of our variables and constants together so we have a C. And a to see on the left side those combined to give us three C. A positive and a -2 on the left side. Those will combine to give us syrup So that equals r. five c plus six. Now I'm going to move my 5C over so you can get see on one side you do that -5 C -5 C. That will turn into -2 C equals six. Finally we can divide both sides by -2. This gets us see on one side and six over negative two which is negative three. So the answer to our solution that satisfies the following conditions as C equals negative three which is our answer as answer C. Okay I hope that we solve the problem. Thank you for watching. Goodbye

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