A rational equation is one that contains fractions. Yes, we will be finding a common denominator that has ‘x’s. But no worries! Together we will use a process that will help us solve rational equations every time!

## Rational Equations

A rational equation is an equation that contains fractions with *x*s in the numerator, denominator or both. Here is an example of a rational equation: (4 / (*x* + 1)) – (3 / (*x* – 1)) = -2 / (*x*^2 – 1).

Let’s think back for a moment about solving an equation with a fraction. 1/3 *x* = 8. We think of the 3 in the denominator as being a prisoner, and we want to release it. To set the 3 free, we multiply both sides of the equation by 3. Think of it as 3 letting both sides of the equation know he’s leaving. 3 (1/3 *x*) = 8 (3).

This process freed our denominator and got rid of the fraction – *x* = 24. It is also the process we use to solve rational equations with one extra step. In rational equations, sometimes our solution may look good, but they carry a virus; that is, they won’t work in our equation. These are called extraneous solutions. The steps to solve a rational equation are:

- Find the common denominator.
- Multiply everything by the common denominator.
- Simplify.
- Check the answer(s) to make sure there isn’t an extraneous solution.

Let’s solve a couple together.

## Example #1

Example number one: solve. Remember to check for extraneous solutions. (3 / (*x* + 3)) + (4 / (*x* – 2)) = 2 / (*x* + 3).

Our first step is to figure out the terms that need to be released from the denominators. I look at 3 / (*x* + 3). I write down (*x* + 3) as one of my common denominators. I look at 4 / (*x* – 2). I write down (*x* – 2) as another part of my common denominator. I look at 2 / (*x* + 3). Since I already have (*x* + 3) written in my denominator, I don’t need to duplicate it.

Next, we multiply everything by our common denominator – (*x*+3)(*x*-2). This is how that will look: ((3(*x* + 3)(*x* – 2)) / (*x* + 3)) + ((4(*x* + 3)(*x* – 2)) / (*x* – 2)) = (2(*x* + 3)(*x* – 2)) / (*x* + 3))

It isn’t easy for the denominators to be released; there is a battle, and like terms in the numerator and denominator get canceled (or slashed). Slash (or cancel) all of the (*x* + 3)s and (*x* – 2)s in the denominator and numerator. Our new equation looks like: 3(*x* – 2) + 4(*x* + 3) = 2(*x* – 2).

Distribute to simplify: (3*x* – 6) + (4*x* + 12) = 2*x* – 4. Collect like terms and solve. 3*x* + 4*x* = 7*x*, -6 + 12 = 6. We end up with 7*x* + 6 = 2*x* – 4.

Subtract 2*x* from both sides: 7*x* – 2*x* = 5*x*. Subtracting from the other side just cancels out the 2*x*, and we get 5*x* + 6 = -4. Subtract 6 from both sides: -4 – 6 = -10. Again, subtracting 6 will cancel out the +6, so we end up with 5*x* = – 10. Divide by 5 on both sides, and we cancel out the 5 and give us *x* = – 2. It turns out *x* = – 2.

The reason we check our answers is that sometimes we get a virus, or, in math terms, extraneous solutions. To check, I replace all the *x*s with -2: (3 / (-2 + 3)) + (4 / (-2 – 2)) = (2 / (-2 + 3)). Let’s simplify: (3 / 1) + (4 / -4) = (2 / 1). Since 3 + -1 = 2 is true, *x* = – 2 is the solution!

## Example #2

Example number two: solve. Remember to check for extraneous solutions. (4 / (*x* + 1)) – (3 / (*x* – 1)) = -2 / (*x*^2 – 1).

First we need to release our denominators. To release our denominators, we write down every denominator we see. I have found the easiest way to do this is to first factor, if needed, then list the factors. *x*^2 – 1 = (*x* + 1)(*x* – 1).

Our new equation looks like this: (4 / (*x* + 1)) – (3 / (*x* – 1)) = -2 / (*x* + 1)(*x* – 1).

I look at 4 / (*x* + 1). I write down (*x* + 1) as one of my common denominators. I look at 3 / (*x* – 1). I write down (*x* – 1) as another part of my common denominator. I look at -2 / (*x* + 1)(*x* – 1). Since I already have those written in my denominator, I don’t need to duplicate them. So my common denominator turns out to be (*x* + 1)(*x* – 1).

Kathryn, why aren’t we using the factors of *x*^2 – 1? Great question! We already have (*x* + 1) and (*x* – 1) being released. We don’t need to do it twice.

Now we multiply each part of the equation by the common denominator – (*x* + 1)(*x* – 1). Think of this as the key to the prison: (4 (*x* + 1)(*x* -1) / (*x* + 1)) – (3 (*x* + 1) (*x* – 1) / (*x* – 1)) = -2 (*x* + 1)(*x* – 1) / (*x* + 1)(*x* – 1).

It isn’t easy for the denominators to be released; there is a battle, and like terms get canceled (or slashed)! Slash (or cancel) all of the (*x* + 1)s and (*x* – 1)s in the denominator and numerator. This leaves us with 4(*x* – 1) – 3 (*x* + 1) = -2.

Now we need to solve for *x*. Distribute 4 into (*x* – 1) and -3 into (*x* + 1). (4*x* – 4) – (3*x* – 3) = -2. Collect like terms: *x* – 7 = – 2. Add 7 to both sides of the equal sign: *x* = 5.

It looks like our answer is 5, but we need to double-check. I replace all the *x*s with 5 and simplify. It turns out 5 works, and it is the solution to our equation. And so our solution checks!

## Lesson Summary

The steps to solving a rational equation are:

- Find the common denominator.
- Multiply everything by the common denominator.
- Simplify.
- Check the answer(s) to make sure there isn’t an extraneous solution.