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

Question

Accepted Answer

Hey everyone welcome back in this problem for the following equation we're asked to solve for X. And the equation we're given is X plus 6/3, X is equal to x minus 2/6. So rewriting what we're given X plus 6/3 X is equal to x minus 2/6. Now, when we're given an equation like this, what we want to do is we want to try to simplify it, we want to try to combine these terms into one expression so that we can see if it's something more familiar to solve. Maybe it's maybe it's a quadratic function, Maybe it's the equation of a circle. We want to kind of combine this, simplify it and see if it's something that we recognize that we know how to solve for. Now when we're looking at this one, okay, we can multiply both sides by three X and we can multiply both sides by six. This is like your cross multiplication until we get X plus six times six on the left side And we get three x times x -2 on the right hand side. Now you see we have this constant term of six on the left, we have this constant term of three on the right hand side. Let's go ahead and simplify right off the bat. We divide both sides by three. The left hand side is going to give us x plus six times two. And on the right hand side we're just left with X times x minus two. Now, if you didn't see that off the bat, you could go ahead and expand first and you would be able to simplify those numbers later on. But if you do notice that right away, if you can do that division while it's simple, then your numbers are gonna be smaller when you expand and can be a little bit easier to work with. Now before we keep going, we need to be careful what our original equation was. So we had an original equation where we divided by three X. If X is zero then the denominator three X is going to be equal to zero and we're going to be dividing by zero, which we know we can't do because the equation will be undefined. So now that we've rearranged this equation so that there's no denominator, we don't have that information. Okay. So what we need to do is add a restriction that X cannot be equal to zero and we're gonna carry that restriction with us all the way through to the bottom of the problem. So that when we get our possible answer choices, we can double check that they don't interfere with one of those restrictions. Right? So always if you're multiplying terms up, if you're getting rid of denominators always double check if there are any restrictions in the original equation or the previous equation before you continue on. Okay, If there are, write them down and keep them handy. So you can double check at the end that your solutions that you're finding our true solutions. Now that we've got our restrictions in check. We've got this expression, let's go ahead and expand on the left hand side, we're going to have two X plus 12. And on the right hand side we have X squared minus two X. And you'll notice that we have X squared terms, we have X terms and we have a constant term. So this is gonna be the equation of a problem. Okay, this is a quadratic equation, a polynomial of degree two, whatever you wanna call it, Okay, we know how to solve for X. It's like finding the roots, it's like solving for the zeros and we know how to do that. So let's move everything over to the right hand side, we're gonna have zero is equal to x squared, we have minus two X. And then we're gonna have minus two X from the left hand side. So we get minus four X And then -12. Now, in order to solve for x, we want to go ahead and factor so you can factor in whichever way you like. And the factors here are going to be x minus six and X plus two. And if you ever get stuck on a test or working through a problem and you aren't able to factor, you're having trouble finding those factors. Remember that, you can always try to use the quadratic formula to find those factors to find the solutions, those X values of this quadratic. Okay, so if you ever get stuck factoring you can always use that method instead. Now we have two things multiplied together that are equal to zero. So if either of these factors are equal to zero then the entire right hand side is going to be zero and that's going to satisfy the equation. So the solutions are going to come from when X minus six is zero, which is going to happen when X is equal to six and when X plus two is zero, which is going to happen when X is equal to negative two. So we have these two X values six and negative two. And we want to double check with our restriction are restriction was that X can't be zero are two solutions that we found are not zero. So that's good. They aren't part of our restrictions so they are true solutions. And so our solutions are X equals six and X equals negative two. And that's going to correspond with answer. C Thanks everyone for watching. I hope this video helped you in the next one.

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

Key Concepts

Solving Rational Equations

Cross-Multiplication

Checking for Extraneous Solutions

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