Hey, everyone. We've solved a bunch of different linear equations, but now we're going to look at another type of equation called a rational equation. Our goal here is still the same. We want to find some value for x that will make our equation true, but you might be looking at this equation and not really be sure where to start. But don't worry. We're going to take this rational equation, and we're going to turn it into a linear equation, which we already know how to solve. So let's get started.
A rational equation is an equation that has a variable like x in the denominator or the bottom of a fraction. Looking at my example here, I have 1x-1 equals 12. So this x that I have in my denominator tells me that I'm dealing with a rational equation. As I mentioned before, we can solve a rational equation by turning it into a linear equation. It's really important about rational equations that our solution cannot be any value that's going to make a denominator 0. In my example here, x minus 1 can never be equal to 0. If I were to have solved this example and got the solution x equals 1, and then went to plug that back in, I would get 11-1 which would be 1 over 0. This is definitely not a fraction I want to be dealing with. Because that denominator is 0, I want to avoid this at all costs. So this actually tells me that my answer, my solution cannot be 1. The fact that x cannot be 1 here is what's called the restriction.
So let's go ahead and jump into an example and look at the steps we need to take to solve a rational equation. Here I have xx-1 is equal to 76. My very first step here is going to be to determine my restriction. I will determine what x can't be before we determine what x is, by setting our denominator equal to 0. We only need to do this with denominators that have a variable in them. I am going to take x minus 1 and set it equal to 0. If I solve that, by adding 1 to both sides, I am left with x equals 1. So this tells me that if x was equal to 1, that would make my denominator 0, which is exactly what I don't want, so this is my restriction, meaning x cannot be equal to 1.
Our second step is going to be to multiply by our LCD, our least common denominator, in order to get rid of those fractions. This is exactly what's going to take us back to a linear equation, which is really familiar to us. Since the two denominators here do not have any common multiples, that tells me that my least common denominator is just the product of these. So to eliminate these fractions, I am going to multiply by 6 times (x minus 1). Remember, when we're eliminating fractions, when we're multiplying by our LCD, we need to make sure and distribute it to every single term in our equation. So let's go ahead and expand this. I get 6∗x-1∗xx-1 and that's equal to 6∗x-1∗76. On my left side here, x minus 1 is going to cancel, which is great because I am just left with 6x. And that is equal to on this side, my sixes are going to cancel, so I'm left with 7 times (x minus 1).
Our third step is simply to solve that linear equation. Distribute the 7 into my parenthesis. I am left with 6x equals. This distributed 7 is going to give me 7x minus 7. I will move all my x terms to one side, all my constants to the other. I want to move this 7 over here since my constant is already on that right side. So to do that, I need to subtract 7x from both sides. It will cancel, and I am left with -1x is equal to -7. My last step in solving this linear equation is going to be to isolate x, which I can do by dividing by -1. I am left with x equals 7, and this is my solution.
Step 4 is to check our solution with our restriction. We found our restriction that x cannot be equal to 1. And my solution x equals 7, which is definitely not 1, so we're good to go, and our solution is x equals 7. That's all for this one. I'll see you in the next one.
