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.(x−6)/(x+5) = (x−3)/(x+1)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to find a solution to the following rational equation. Otherwise choose no solution. Our rational equation is on the left hand side, in the numerator, we have the difference of X minus four and that's divided by the sum of X plus one. And that's all equal to, on the right hand side in the numerator, we have the difference of X minus two and that's all divided by the sum of X plus seven. Our answer choices are answer choice. A no solution. Answer choice B X equals seven halves, answer choice C X equals 11 halves and answer choice D X equals 13 halves. All right. So we have two fractions that are equal to each other. And in order to begin to solve for X, we want to get rid of the fractions. So we can do that by cross multiplying, we can multiply the numerator from the left hand side by the denominator from the right hand side. And likewise multiply the numerator from the right hand side by the denominator on the left hand side. So let's do that. We'll have the difference of X minus four multiplied by the sum of X plus seven. And that's still equal to. Now, the difference of X minus two multiplied by the sum of X plus one. Now we can foil both sides of this equation. So on the left are firsts are X multiplied by X which is X squared, our outsides X multiplied by a positive seven which is plus seven X, our insides negative four multiplied by X will have minus four X. And then our last a negative four multiplied by seven will be a negative 28. And that's all equal to. Now, we can foil out the right hand side. Our firsts X multiplied by X X squared and then our outsides X multiplied by a positive one will have plus one X which is just plus X and then our insides negative two multiplied by X which will give us a minus two X and our last negative two multiplied by one will have minus two. All right. Now, we can combine the outside and inside like terms on each side. So on the left, we'll bring down that X squared, then we have plus seven X minus four X which will leave us with plus three X and then bring down our minus 28. That's all equal to. On the right, we can bring down X squared. We're taking our one positive X minus two X which will be minus one X or just minus X and bring down this minus two. All right, we want to now start bringing all of our XS together. We can start to bring them over to the left hand side of the equation. So I will subtract X squared from the right hand side to make that equal to zero for our X squared. But taking away an X squared from the left hand side as well will leave us with no X squared on that side, either X squared minus X side squared will be zero. Now let's move over this negative one X, we'll add X to the right hand side and add one more X to the left hand side as well. And now to bring the constant terms over to the right hand side of the equation, we can add 28 to both sides because on the left, negative 28 plus 28 will be zero. So now let's simplify what we've got left. So on the left, we have three X plus X that will be a positive four X and then on the right, that negative X plus X, that one was zero and the negative two plus will give us a positive 26. Now to get X by itself, it's being multiplied by the four. The opposite of multiplication is division. So we're going to divide both sides by four. On the left four, divided by four is one. So we have one X which is just X and we have X by itself, which is what we wanted. And on the right hand side, 26 divided by four. Well, those are both divisible by two. We want to keep it as a fraction to keep it as accurate as possible. In our numerator 26 divided by two is 13. And in our denominator four, divided by two is two. And so X is equal to 13 halves, we look at our answer choices and that matches with answer choice. D well done. We'll catch you on the next one.

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In Exercises 1–34, solve each rational equation. If an equation has no solution, so state.(x−6)/(x+5) = (x−3)/(x+1)

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