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How to Solve a Rational Equation

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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!

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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:

  1. Find the common denominator.

  2. Multiply everything by the common denominator.

  3. Simplify.

  4. 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).

In example #1, the first step is finding the common denominator. Finding the common denominator

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.

Like terms are cancelled out or slashed in the second example. Cancelling Like Terms Rational Equation

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:

  1. Find the common denominator.

  2. Multiply everything by the common denominator.

  3. Simplify.

  4. Check the answer(s) to make sure there isn’t an extraneous solution.

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