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

Question

Accepted Answer

Hey everyone welcome back for the following equation. We're asked to solve for X. And the equation we're given is X over X minus 11 is equal to 11 over X minus 11 plus 11. A lot of 11's there. Alright, so let's write what we're given. Okay? And then we're gonna get started. So when we're given an equation like this, what we want to do is we want to go ahead and combine everything into one term if we can Okay, so we have three separate terms right here, we want to combine these into one expression K one kind of rational expression set that have that equal to zero on one side and then we're going to be able to solve for X much more easily. Okay, so in order to do that we need a common denominator. Now you'll see in the term on the left hand side, we have a denominator x minus 11 In our second term we have a denominator X -11. So let's go ahead and get that final term that plus 11 to have that denominator as well. So we have X over X minus 11 is equal to 11 over x minus 11. Now we want this last term to be over x minus 11 as well in order to do that, we have to multiply by x minus 11 in the denominator. If we do it in the denominator we have to do in the numerator as well. Okay because then we have x minus 11 over x minus 11, it's like we're multiplying by one, which doesn't change our equation. Alright, so we get 11 times X -11 over X -11 for that last term. Now we have a common denominator so we can go ahead and move everything to one side, combine all of these terms into one. So let's move everything to the right hand side. So on the left hand side we're gonna get zero on the right hand side, we're gonna have 11 plus 11 times X minus 11. That's gonna give us 11 x minus 121. And then we're gonna have minus X from moving the term from the left hand side over to the right hand side and all of this is going to be divided by our common denominator, X minus 11. Alright, so let's just simplify that numerator. We have zero is equal to we have 11 x minus X. Which is going to give us 10 X. And then we have minus 121 plus 11. That's going to give us minus 110 Divided by X -11. Alright, now when is this going to be equal to zero? Well this is going to be equal to zero. If the numerator is equal to zero we can multiply both sides by X -11. On the left hand side, we have zero. And so we just get zero is equal to 10 X -110 But we have to be careful okay because we've gotten rid of the denominator here and we don't have the information but we know that we had originally in our equation a denominator of X -11. If the denominator is equal to zero, we're going to be dividing by zero which we know we can't do the function or the equation will be undefined. So that can't be a solution. So if X minus 11 is equal to zero, which means that X is equal to 11, No solution. OK. So we have this restriction where X cannot be equal to 11 and it's important to carry that restriction with you for the rest of the problem. Okay, when you get those potential solutions at the end you're gonna want to check back and make sure they aren't a restriction. Alright, so we have 10 X minus 1. 10 is equal to zero. Okay, we know how to solve this. This is just a simple equation. We have 10. X is equal to 110 which gives us an X is equal to 11. Okay, So great. We found X is equal to 11. That's our solution. No. Stop. Okay, think about it. We have this potential solution. X is equal to 11. But remember we said we always have to go back and check our restrictions and our restriction is that X cannot be 11. Okay, if X is equal to 11 then the denominator of our original equation is equal to zero. Okay, so we actually end up with no solution here. Okay, so it's always important to write those restrictions and go back and double check that the potential solution you found is not one of those restrictions. Okay? Because it can change your answer drastically. Okay. Changes from having a solution of a particular value to having no solution. Alright? So if we go back up to our answer choices, we found that we have no solution, so we have answered. D Thanks everyone for watching. I hope this video helped see you in the next one.

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

Key Concepts

Rational Equations

Cross Multiplication

Extraneous Solutions

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