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.(7x−4)/5x = 9/5 − 4/x

Accepted Answer

Welcome back. I am so glad you're here. We are asked to find the 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 13 X minus 11. That's divided by the denominator nine X. That is equal to, on the right hand side, we have 19 12th minus four divided by X. Our answer choices are answer choice. A X equals 20 answer choice B X equals 25 answer choice C X equals and answer choice D no solution. All right, we have three fractions here. And because we have an equal sign in the middle, in order to solve for X, we want to get rid of these fractions, we will do that by multiplying all three of the fractions by the lowest common denominator. So looking at our denominators nine X 12 and X, what's the lowest common denominator? Well, for the coefficient and the constant terms for nine and 12, the lowest common denominator is 36. And then both of these Xes are to the first degree. So it's just going to be 36 X now we're taking each of the three fractions and multiplying their numerators by 36 X. And what we'll do next is we look between the 36 X and the denominator and see how they reduce each other. So for the first one, on the left hand side, the 36 the nine, those are going to reduce because 36 divided by nine is a positive four. So we will take the 36 that becomes a four and our s those cancel each other out. So now we will have a four multiplied by the difference of 13 X minus 11. Now, on the right hand side of the equal sign, we take a look at our 36 X and divide that by 12. Well, 36 divided by 12 is a positive three and then that X gets to stay because there's no X in the denominator to cancel that out. So we have a three X being multiplied by the 19. And now for the last fraction, we have 36 X divided by X. This time, the X is canceled out and that 36 remains. So we can bring down this minus and then we have a 36 multiplied by what's left of the fraction, which is just four. Now, we need to distribute what is in front of all of the sets of parentheses. So on the left hand side, we can take four multiplied by 13 X and negative 4, multiplied by 13 X will give us X four multiplied by negative 11 is in negative 44. That's equal to on the right hand side, three, X multiplied by 19 is 57 X and a negative 36 multiplied by four is a negative 144. Now, it's just a matter of getting X by itself on one side of the equation and all of the other numbers to the other side, I prefer to keep X positive when possible. And the X is on the right hand side are bigger than the X is on the left hand side. So I'm going to subtract 52 X from both sides of the equation because on the left 52 X minus 52 X is zero. And then I am going to add to both sides of the equation to bring the constant terms to the left hand side of the equal sign because on the right, negative 144 plus 144 is zero. So now let's see what we've got left. On the left negative 44 plus 144 is a positive 100 that's equal to on the right 57 X minus 52 X is a positive five X. Now X is being multiplied by five and to undo that multiplication, we will divide. So we take five X divided by five, what we do to one side, we do to the other as well. On the right five X divided by five will be one X or just X, which is exactly what we wanted. We have X by itself ya. And on the left 100 divided by five is a positive 20. And we've solved for X X is equal to 20. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

In Exercises 1–34, solve each rational equation. If an equation has no solution, so state.(7x−4)/5x = 9/5 − 4/x

Key Concepts

Rational Equations

Finding a Common Denominator

Checking for Extraneous Solutions

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