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
3*x*(*x* – 2) = *x*(*x* + 1). This simplifies to 3*x*^{2} – 6*x* = *x*^{2} + *x*, which can be rearranged into 2*x*^{2} – 7*x* = 0. Now we have a polynomial equation, and we know what to do. First, factor: *x*(2*x* – 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 2*x* – 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 3*x*(*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. |