 So let's try to find all solutions to the equation 1 over 5x equals x minus 8 over x squared plus 4x. First, since this is an equation involving some rational expressions, it's possible that there may be some forbidden values. So first, we find the forbidden values of x. And those are things that would make the denominator equal to zero. So since the denominator can't be zero, we require these two denominators, 5x can't be zero, x squared plus 4x can't be zero. And remember we solve an inequality by solving the corresponding equality. If I don't want 5x to be zero, first I try to solve 5x equals zero. So we'll solve, and x equals zero makes 5x equals zero, but we don't want that. So we require x not equal to zero. We have our second denominator, x squared plus 4x, not equal to zero. So we solve the corresponding equality, x squared plus 4x. And here we have an easy factorization since both terms have a factor of x, so we factor. And now we have product equal to zero. So either x is equal to zero, and we'll solve that by doing nothing, really. So 4x plus 4 is equal to zero, which we solve. And again, x equals negative 4 makes the equation true, but we don't want that. So we also require x not be equal to negative 4. Now before we go any further, remember that factored form is best. Having gone through the trouble of factoring x squared plus 4x as x times x plus 4, let's go ahead and make use of that. We'll remember that that is our factorization. Remember, if it's not written down, it didn't happen, so we'll write down our equation with this factored form. So I have a rational expression, 1 over 5x, and another rational expression, x minus 8 over x times x plus 4, and we can eliminate the denominators by multiplying through by those denominators. So let's get rid of this denominator 5x. We'll multiply the left-hand side by 5x, and in order to maintain the equality, we also multiply the right-hand side by 5x. And so this common factor of 5x on the left can be removed, leaving us with 1. On the right-hand side, the denominator has a factor of x, and we're multiplying by a factor of x, so we can remove this common factor as well, leaving us with just that factor of 5. So our left-hand side is 5 times x minus 8 over x plus 4. Well, we still have this denominator x plus 4, so let's multiply both sides by the denominator x plus 4. Over on the left-hand side, we have x plus 4 times 1, and there's not really a whole lot we can do with that. On the right-hand side, the factor of x plus 4 in the denominator, and the factor x plus 4 we're multiplying by are common factors, so they can be removed, and that leaves us with 5 times x minus 8. So we'll expand, and that gives us a new equation. So let's solve that equation. Over on the right-hand side, we have a difference. Over on the left-hand side, we have a sum, and it's easier to get rid of a difference because all we have to do in that case is add. So let's add 40 to both sides. Over on the left-hand side, we have a sum, so we can either subtract x or subtract 44. If we subtract x, that'll eliminate it from the left-hand side, and we already have x on the right, so that'll give us a simpler equation. So we'll subtract x from both sides. And now, over on the right-hand side, we have a product for times x, so we'll get rid of that by dividing by 4, and that gives us our solution, x equal to 11. Now remember we had two forbidden values, x cannot be equal to 0, x cannot be equal to negative 4, but 11 is all right. Since 11 is not a forbidden value, it is in fact the only solution to this equation.