Today we're going to talk about solving rational equations. A rational equation is just an equation with one or more rational expressions. So we're going to start by talking a little bit about the process for solving rational equations. And then we'll do some examples.
So here's an example of a rational equation, 2x squared over x plus 3 plus x over 8 equals 4. Now we know that when we have a fraction, the denominator cannot be equal to 0. So when we're solving rational equations, we need to be on the lookout for extraneous solutions. And an extraneous solution is just an invalid solution to the equation.
We get extraneous solutions when we have a value for our variable that would make the denominator of a fraction equal to 0. So in other words, in this fraction where we have a variable in our denominator, whatever value for x makes this denominator equal to 0 is going to give us an invalid or an extraneous solution. So to find what value of x that would be, we can set this denominator, the expression in the denominator, equal to 0.
So if x plus 3 equals 0, to find out what x is I would subtract 3 from both sides. And I see that x is equal to negative 3. So when x is equal to negative 3, that would give us an invalid or extraneous solution to our rational equation.
So let's talk about how we can find the least common denominator between algebraic fractions. I've got the algebraic fractions x plus 3 over 4x, 5 over x squared, and x over 6. So we can always find a common denominator by simply multiplying all of the denominators together. And this expression would be our common denominator. However, it could be overly complicated than what we need to combine our fractions.
And so we can find a least common denominator in a different way. Now before we do that, we need to make sure that all of our fractions are as simplified as they can be. And they are. There's no common factors in the numerator and the denominator of any fraction that could be canceled out.
So to find my least common denominator, I'm going to start by writing each of my denominators in factored form. So 4x I can write as 2 times 2 times x. x squared I can write as x times x. And 6 I can write as 3 times 2.
Now, my least common denominator is going to be an expression that includes all of the terms from any one denominator-- sorry, all of the factors from any one denominator, but nothing more than it needs to. So if I look at my first denominator, 2 times 2 times x, I know that it has to include all of those factors.
But when I look at my second denominator, I have x times x. I already have an x in my expression, so I only need to multiply by one more x. So now I have two x's for my second denominator.
And for my third denominator with a 3 and the 2, I already have the 2 as a factor. So I just need to multiply by the 3. And then I have both factors that I need for that denominator.
So now I can simplify this. So my least common denominator is going to be 2 times 2 times 3, which is 12. And then x times x gives me x squared, so 12x squared.
So now I can rewrite each of my algebraic fractions with a common denominator of 12x squared. And to do that, I want to multiply by some factor, some term to make the denominator 12x squared. So to make 4x be 12x squared, I first need to multiply the 4 by 3 to get the 12. Then I'll need to multiply x by another x to get x squared. If I multiply by 3x in the denominator, I'll need to do that in the numerator.
For the second fraction, I'll just need to multiply by 12 to make x squared 12x squared. And for my third fraction, I'll have to multiply by 2 to make the 6 to be 12, and then x squared.
So now I can simplify each of these fractions. For my first fraction in the denominator we'll have 12x squared, which is what we wanted. And in our numerator, we need to be careful to multiply this 3x by both terms in the parentheses. So 3x times x would give me 3x squared. And 3x times 3 will give me 9x.
My second fraction, I'll have 60 in the numerator. 5 times 12 is 60. And in my denominator, I'll have the 12x squared. And for my third fraction, this will give me 2x to the third power over my common denominator of 12x squared.
So now let's do an example solving a rational equation. I've got 3x minus 1 over x squared plus 3 over 6 equals x minus 1 over 2x. So I want to solve this rational equation first by finding the least common denominator between each of my algebraic fractions.
So to find the least common denominator, I'm going to start by writing each of my denominators in factored form. So x squared could be written as x times x. 6 can be rewritten as 2 times 3. And 2x can be rewritten as 2 times x.
So then my least common denominator I know needs to include all of the factors from each denominator, but no more. So in my first denominator I have x times x, so I need to include that. For my second denominator, I have 2 times 3, so I need to include that. For my third denominator, I have 2 times x. I already have 2 and an x, so I don't need to write them again.
So then I can just simplify this. 2 times 3 is going to give me 6. And x times x is x squared. So I found my least common denominator is 6x squared.
So now I can multiply each of my denominators by a turn to make them into 6x squared. And then I'll have to multiply also in the numerator too, so that I'm not actually changing the value of the fraction. So to make x squared be 6x squared, I need to multiply by 6. And so if I do that in the denominator, I'll need to multiply it in the numerator. And I'm going to multiply by six to both terms in the numerator.
For my second fraction, to make 6 into 6x squared I'll have to multiply by x squared. And to make 2x into 6x squared I'll have to multiply by 3 and x.
So now simplifying each fraction, my first fraction, multiplying the 6 by both 3x and negative 1, 6 times 3x will give me 18x. And 6 times negative 1 will give me negative 6 over my denominator of 6x squared. My second fraction will become 3x squared over 6x squared. And here I'll again multiply the 3x by both things in parentheses, so this will become 3x squared minus 3x over 6x squared.
Now that my denominators are all the same, I can in essence ignore them and write an equation using only the terms in my numerators. So I'll have 18x minus 6 plus 3x squared equals 3x squared minus 3x. So now to solve this equation for x, I'm going to start by looking for any like terms.
I see here that I have a positive 3x squared on both sides of my equations so these two terms will just cancel out. And I'll have 18x minus 6 is equal to negative 3x. The minus needs to stay with the 3x.
Then I'm going to combine my x terms. I'm going to subtract 18x on both sides. Here this will cancel. And I'll have negative 6 is equal to negative 21x. And then dividing both sides by negative 21, I find that x is equal to negative 6 over negative 21, which reduces to 2/7.
So I found that my solution for x is 2/7. But we need to check for any extraneous solutions. And we do that by making sure that this value for x does not make the denominator of any of my fractions equal to 0.
So 2/7 substituted for my x value here will not give me 0. And 2/7 substituted for my x value here will not give me 0. And so I've checked all of my fractions that have a variable in the denominator. And I found that this is a valid solution.
Let's go over our key points from today. To solve a rational equation, rewrite each fraction in the equation with a common denominator. Then, because the denominators are the same, you can write and solve an equation using just the numerators of each fraction. Sometimes rational equations have extraneous solutions. If a potential solution makes the denominator of a rational expression equal to 0, then the solution is extraneous.
So I hope that these key points and examples helped you understand a little bit more about solving rational equations. Keep using your notes. And keep on practicing. And soon you'll be a pro. Thanks for watching.
an equation with at least one rational expression