Solve each equation. See Example 1. (4x+3)/4 - 2x/(x+1) = x

Question

Accepted Answer

Hey everyone in this problem for the following equation we're asked to solve for X. And the equation we're given is five X plus 7/5 minus five X over five X plus 11 is equal to X. So we're just going to rewrite what we're given now when we have an equation like this and we're asked to solve for X. What we wanna do is we want to try to combine these three terms were given into one term. Okay we want to have one kind of rational expression on one side and it equal to zero on the other side. If we do that it's gonna be more straightforward to solve for X from that point. Okay so how can we combine these things? So the first thing we need to do is to have a common denominator. Okay so on the left hand turn we have a denominator of five, the second term we have the denominator of five X plus 11. And on the right hand side we have just a denominator of one. So our common denominator okay is going to be five times five X plus 11. Okay we find this in the same way that we would we were dealing with fractions with just numbers. Alright, so for our first we have five X plus seven we're gonna divide it by five and we want to multiply the denominator by five X plus 11. So that we have that common denominator. If we multiply the denominator by that. And we also need to multiply the numerator. If we do that then they were multiplying by five X plus 11/5 X plus 11. Which is like multiplying by one. Which doesn't change our equation. That's what we want. Now the next term we have five X. Okay and in the denominator here we have five X plus 11. So we're missing that term of five. We want to multiply the denominator by five which means we also multiply the numerator by five so that we're multiplying by one. Now on the right hand side we have X. Okay and this is missing both of those terms from our denominator. So we need to multiply this by five and we need to multiply by five x. Plus 11. Okay in both the numerator and the denominator in order to get that common denominator without changing our equation. Alright now that we have a common denominator we can move everything over to one side and combine these three separate terms into one. So on the left hand side we have five X plus seven times five X plus 11. This is going to give us 25 x squared plus 90 X plus 77. This second term we have five X. Times five which is gonna give us 25 X. And we're subtracting so we get minus 25 X. And then we're gonna move this term from the right hand side over. Okay So we have X times five times five X plus 11. This is gonna be 25 X squared plus 55 X. And we're subtracting all of it. So we get minus 25 X squared minus 55 X. All over our common denominator of five times five X plus 11. And this is equal to zero. All right, let's simplify the numerator. See what we got. So for our X squared terms we have 25 X squared minus 25 X squared. So those two are going to add to zero. We have 90 X minus 25 X minus 55 X. That's gonna be 90 X minus 80 X. Which gives us 10 X. And then we have 77. So we have 10 X plus 77 over our denominator of five times five X plus 11. And this is going to be equal to zero. Now, in order for this fraction to be equal to zero we need the numerator will be equal to zero. If the numerator is equal to 00 divided by anything is going to be zero. So what we need to do is find the X values such a 10 X plus 77 is equal to zero. However, we have to be careful in our original function, we had this denominator of five X plus 11. If the dominator is zero then our equation divides by zero, which we know we can't do. Okay, our equation will be undefined which means we won't have a solution. So we need to add in a restriction on X. Okay, if this denominator zero now the denominator will be zero. If five X plus 11 is equal to zero or if X is equal to negative 11/5. So we have this restriction where X cannot be negative 11/5 and we're going to carry that through for the rest of the problem and once we find potential solutions, double check that we don't have a restricted one. Alright, so we have 10 X plus 77 equals zero. If we solve this we're gonna have X is equal to negative 77/10. Okay, this is not equal to negative 11/5. This is not one of our restrictions. And so that is a valid solution. And so our solution is going to be X equals negative 77/10. And if we go back up to our answer choices, we see that that is going to correspond with answer choice. A Thanks everyone for watching. I hope this video helped see you in the next one.

Solve each equation. See Example 1. (4x+3)/4 - 2x/(x+1) = x

Key Concepts

Solving Rational Equations

Finding the Least Common Denominator (LCD)

Checking for Extraneous Solutions

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