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 =12. This x that I have in my denominator tells me that I'm dealing with a rational equation. As I said before, we can solve a rational equation by actually turning it into a linear equation. It's really important about rational equations that our solution, whatever value we get for x, cannot be any value that's going to make a denominator 0. In my example here, this x - 1 can never be equal to 0. If I were to have solved this example and I got the solution x = 1, and then I went to plug that back in, I would get 11-1, which would be 1 over 0. This is definitely not a fraction that 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.
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 = 76. My very first step here is going to determine my restriction. We're going to determine what x can't be by setting our denominator equal to 0. If I solve that, I'm going to add 1 to both sides. I am left with x = 1. This tells me that if x was equal to 1, that would make my denominator 0. That's exactly what I don't want, so this is my restriction. So x cannot be equal to 1.
Now, let's figure out what x is by taking our other steps. Our second step is going to multiply by our LCD, our least common denominator, to get rid of those fractions. This is exactly what's going to take us back to a linear equation, which is familiar to us. Now let's go ahead and expand this. So I get 6 times x-1 times my first term, xx-1. And that's equal to 6 times x-1, my LCD again, times my second term, 76. On my left side here, my x-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-1. I just moved that 7 over to the front just to make it a little bit easier to deal with. We've completed step 2, and you might have noticed that now we just have a linear equation. So our third step is simply to solve that linear equation. The first thing I need to do here is to distribute this 7 into my parenthesis. My left side is going to remain the same, so I'm just left with 6x = 7x - 7.
To isolate x, I need to subtract 7x from both sides. It will then cancel, and I am left with -1x = -7. Now my very last step in solving this linear equation is to divide by negative one. My negative one will cancel, and I am left with x = -7-1. It gives me positive 7, and this is my solution. Step 3 is done, but we still have a final step, step 4, which is going to be to check our solution with our restriction.
Step 1, we found our restriction. We're going to check our solution and make sure that it's not that. So my restriction here was that x cannot be equal to 1. And for my solution, I got x = 7, which is definitely not 1, so we're good to go, and our solution is x = 7. That's all for this one. I'll see you in the next one.
