In Exercises 1–34, solve each rational equation. If an equation h...

Question

In Exercises 1–34, solve each rational equation. If an equation has no solution, so state.

1 − 4/(x+7) = 5/(x+7)

Accepted Answer

Hello, today we're gonna be solving the following rational equation. So what we are given is four minus over X plus three is equal to six over X plus three. Now, for simplicity, we're gonna go ahead and represent the constant four as a rational expression and we can do that by putting it over one. So the easiest way to approach this problem is to go ahead and eliminate the denominators. And we can do that by finding the lowest common multiple between the three denominators. While the lowest common multiple is going to be the product of all the different terms throughout the denominators. And as we can see here, we have X plus three in the 2nd and 3rd term and we have one in the first term. That means the lowest common multiple is going to be one times X plus three, which is just going to give us X plus three. So what we're going to do now is we're going to multiply each term by X plus three in order to eliminate the denominators. So that is going to give us 4/1, multiplied by X plus 3/1 minus 10 over X plus three, multiplied by X plus 3/1. And that is going to equal to six over X plus three, multiplied by X plus 3/1. Now, what we're going to do is we're going to use cross reduction to get rid of the denominators. In the second term, we have X plus three in both the numerator and denominator. So we can just go ahead and eliminate those terms. And that rule also applies to the last term we have X plus three in both numerator and denominator. So we can just go ahead and eliminate those terms. And in the first term, there's nothing further that we can do to simplify. So we're just gonna multiply four with X plus three that is going to leave us with four times X plus three minus 10 is equal to six. Now, what we're going to do is we're going to open up the parentheses. And in order to do that, we're going to distribute four to X plus three that is going to give us four X plus 12 minus 10 is equal to six and 12 minus 10 is going to give us two. So we have four, X plus two is equal to six. Now, what we want to go ahead and do is sol for X. So we're going to subtract two to both sides of the equation and that is going to leave us with four, X is equal to four And finally, if we div divide both sides of the equation by four, that is going to leave us with X is equal to one which is going to be the solution to the given equation. And with that being said, the answer to this problem is going to be D. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 1–34, solve each rational equation. If an equation has no solution, so state. 1 − 4/(x+7) = 5/(x+7)

Key Concepts

Rational Equations

Finding a Common Denominator

Checking for Extraneous Solutions

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