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 multiply both sides of the equation by (x + 2) to find
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 7/2 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 the equivalent equation
Since the denominators are the same, we compare numerators, which means we need to solve the equation
3x(x – 2) = x(x + 1)
in 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.