Hi, and welcome. My name is Anthony Varela. And today we're going to solve rational equations.
So we'll start off by talking about what rational equations are and how they involve rational expressions or algebraic fractions. We'll discuss how finding the lowest common denominator will really help us simplify and then eventually solve our equation. And we'll talk about extraneous solutions.
So let's start with a definition of a rational equation. Well, it's an equation with at least one rational expression. So let's start off with a simple example. Here we have 3/x plus x plus 2, over x, equals 5/x.
So we see in its equation, we have two quantities that are set equal to each other. And we have at least one rational expression. Now I said that this was a simple example.
What makes it so simple? Well, notice our denominator is the same in all of our terms. We have something over x for every single term in our equation.
So really, at least for now, I can in a sense ignore my denominator and just solve this by focusing on my numerators. Another way to think about this it would be multiplying the entire equation by that denominator, x. So I would get 3 plus x plus 2 equals 5.
And this is an equation that I feel much more comfortable solving. I know that this is 5 plus x equals 5. So I solve this as x equals 0.
But here's the tricky thing with rational equations. Now our denominator is not allowed to equal 0. That's with any fraction, whether it's algebraic or numeric. We can't have our denominator equal 0.
And in this case, then, the only x value that's not allowed is x equals 0. But that's the solution I got when I solve this equation. So that's called an extraneous solution.
So it is a solution that you think might be valid because I made no algebraic errors when I was finding this value for x. But when I went back to my original equation, I noticed that that was a domain restriction. x is not allowed to be 0. So while I don't think that I quite did anything wrong here, it's not a valid solution. It's extraneous.
So I mentioned that finding the lowest common denominator of rational expressions is really going to help us solve our equation. So we're going to practice finding the lowest common denominator before we move on to solving another equation.
So we're going to find the lowest common denominator reaching all of these fractions. Now one common strategy to find just any old common denominator would be to multiply all of our denominators together. That would certainly make them common. We'd have to adjust the numerator.
But out with algebraic fractions, that strategy can get a bit messy. So this is how we're going to find the lowest common denominator.
The first step is to factor each denominator completely, as much as you can. So taking a look at our first fraction, 3x in the denominator, I'm going to factor that. Well, that's 3 times x.
Taking a look at our middle fraction, that's actually written out in factored form. One factor is x plus 1. The other factor is x plus 3. So those are our two factors for that denominator. And in our last fraction, we just have one factor of x.
So now that we've factored each denominator as much as we can, we're going to look at the greatest number of times each factor is used. So what I mean by that is taking a look at 3.
And I'm looking at how many times is 3 used as a factor here. Well, once. And how many times is it used in these other factorizations? Well, not at all.
So the greatest number of times this is used is once. So we're going to then multiply that together to be part of our lowest common denominator. So I'm going to write down 3. That's going to be part of our lowest common denominator.
Next I'm going to look at x. Well, this is used once here. It's used zero times here as a factor. Remember, the factor is x plus 1 and x plus 3. So x is not a factor here.
It's a factor here once. So the greatest number of times it's used is once. So I'm going to include x in my lowest common denominator just one time.
And now I have x plus 1. I don't see that anywhere else in my factorizations. So I'm going to pull that down into my lowest common denominator. Same thing with x plus 3. And remember, we already accounted for our one factor of x that we're going to include in our lowest common denominator. So this is our lowest common denominator.
So what that means now, I'm just writing my fractions vertically now. We want to rewrite each fractions so that their denominators are our common denominator. 3x times x plus 1 times x plus 3. Looks a little bit messy, but this is what we're going to do.
With our first fraction, we're just going to bring this over. We have a 2 minus x. And now what factors do we have to multiply into our numerator?
Well, taking a look at 3x, it already appears in our denominator here. So we need to tack on an x plus 1 and an x plus 3. So these two fractions are equivalent.
Taking a look at our middle fraction, we're going to bring over that one factor in the numerator. And now take a look at what factors need to be multiplied by that numerator. So we already have the x plus 1 and the x plus 3. So we need to multiply our numerator by 3x. So now these two fractions are equivalent.
Lastly, we have 2x minus 1 over x. So we're going to bring our numerator over. And now what do we have to multiply into that numerator?
Well, we already have a factor of x. So we're just going to bring in this factor of 3, then. And then we have x plus 1 and x plus 3. So now these two fractions are equivalent.
Now that we know what to look out for denominators being equal to 0, and we have a strategy for finding the lowest common denominator, let's go ahead and solve this rational equation. We have 3 over x squared plus 1/x equals 4. So our first step is to rewrite all of these fractions so that they have a common denominator-- more specifically, the lowest common denominator.
So I'm going to factor out my denominators. Well, I have x squared. I'm going to factor as x times x. And x is just x.
This has a denominator of 1. I don't really need to write that. So we'll come back to that later though.
So now taking a look at what my lowest common denominator will be, I see x is used two times in this factorization. It's used once here. So the greatest number of times it's used is twice.
So that's going to be part of our lowest common denominator. So our LCD is x squared. Two factors of x.
So this is already written with our common denominator, 3 over x squared. How am I going to write 1/x with a denominator of x squared? We'll just have to multiply the numerator by x. So x over x squared is equivalent to 1/x.
4 is 4/1. So if my denominator needs to be x squared, the numerator is going to be 4x squared. So 4x squared over x squared is equivalent to 4.
So now I have my equation written so that every single term has a common denominator. So I'm going to focus then on my numerators. And I'm going to solve 3 plus x equals 4x squared.
I can rewrite this as a quadratic equation set equal to 0 and use any number of strategies-- factoring, completing the square, using the quadratic formula. But first, I need to express this as 0. So I'm going to subtract a 3 and subtract an x.
So really, this should say 0 equals 4x squared minus x minus 3. And if I use the quadratic formula where a is 4, b is negative 1, and c is negative 3, I would get x values of negative 0.75 and 1. Those are my solutions to this equation right here.
Now the only thing that's left is to check to make sure that these are not extraneous solutions. So bringing back my original rational equation, I can double check and see, OK, well, the x values here just can't be 0. And I didn't find x equals 0 as part of my solution. So both of these x values are valid solutions to our rational equation.
So let's review solving rational equations. Well, we can't have a denominator equal to 0. So if that's the case, we have an extraneous solution. It's restricted from the domain, even though you might think you've done everything right.
Finding the lowest common denominator, it will be your key to solving rational equations. Factor each denominator, take a look at the greatest number of times each factor is used, multiply those together-- that's your lowest common denominator.
Once you have a common denominator, just solve using those numerators. But make sure to confirm that your solution is not extraneous.
Well, thanks for watching this tutorial on solving rational equations. Hope to see you next time.