Equations Involving Rational Expressions Examples

Solve the equation .

The first thing we do is look for "bad" values. The "bad" values for this equation are x = -1, which makes the left-hand side of the equation undefined, and x = 2, which makes the right-hand side undefined. These values are definitely from the wrong side of town.

Now we go ahead and solve.

Way 1: Eliminate denominators. Multiply both sides of the equation by (x + 1) and  (x + 2) to find:

3x(x – 2) = x(x + 1)

This simplifies to:

3x² – 6x = x² + x

Then we can rearrange the whole kit 'n' caboodle:

2x² – 7x = 0

Now we have a polynomial equation, and we know what to do. First, factor:

x(2x – 7) = 0

From the factored form we can see that the solutions to the equation are x = 0 and (the solution comes from solving the equation 2x – 7 = 0). Before writing down our final answers, we need to make sure 0 and aren't "bad" values. We should also make sure that they're not seeing each other, so we can be assured they're not having a "bad" romance.

Our two solutions are not, in fact, "bad" answers, so our final answers are:

x = 0,

Way 2: Put the fractions over a common denominator. We multiply the left-hand side of the original equation by and the right-hand side of the original equation by . This gives us an equivalent equation:

Since the denominators are the same, we compare numerators, which means we need to solve this equation:

3x(x – 2) = x(x + 1)

We'll use the same way that we did in Way 1. No way! Yes way. We know that we'll find the same solutions to the equation, which is lovely.

Solve the equation .

First we check for bad values. Factoring the left-hand side of the equation gets us to , which means that -7 and 9 are "bad" values that can't possibly be solutions.

Now we can go ahead and solve this sucker.

Way 1: To eliminate the denominators, multiply both sides of the equation by (x + 7) and (x – 9). Boom:

x² – 2x + 81 = x(x – 9) + x(x + 7)

Then we simplify:

The solutions to this equation are x = 9 and x = -9. However, checking back to see the "bad" values shows us that x = 9 is an extraneous solution and not a real solution to the original equation. Well, well, well, x = 9. The mask is coming off now, isn't it?

The only solution to the original equation is x = -9.

Way 2: The left-hand side of the equation is already a single fraction. To turn the right-hand side into a single fraction, we need to do some addition. Bust out your scientific calculator. On second thought, though, we may be able to do this one manually.

Now that the denominators of the fractions on either side of the = sign are the same, we need the numerators to be the same. We need to solve this equation:

x² – 2x + 81 = 2x² – 2x

This is the same equation we had to solve using Way 1, so we'll find the same answer. If we don't, something may be rotten in the state of Denmark, or whatever state you live in.