 So we can solve a rational equation. If we can rewrite it in the form x over y equals 0, in which case our solution will be numerator equals 0, provided our denominator isn't equal to 0. But this always requires doing a lot of work with fractions. And so you might wonder, is there an easier way? And the answer is, not really, but yes. And so this better way centers around the following idea. In general, if we have a rational equation, we can transform it into a polynomial equation by multiplying by the denominators. And that's because for any expression, x over y, we have x over y times y will just be the numerator x. But there's a catch. Any values that were forbidden before cannot be permitted as solution values. So for example, let's take the rational equation, 3 divided by x plus 2 plus 5 divided by x minus 1. Now if we look at our rational expressions, we see that we require x cannot be negative 2, x cannot be equal to 1, because these would make one of our denominators equal to 0. But as long as we accept that limitation, we can try to simplify this. So remember that for any expression, x over y, x over y times y gives us x alone. So if we multiply everything by x plus 2, we'll eliminate that first fraction. So we have to do the same thing to everything. So if we're going to multiply the first fraction by x plus 2, we'll also have to multiply the second fraction by x plus 2, and also the terms on the right hand side by x plus 2. So in the first fraction, we'll have this common factor of x plus 2 that we can remove, which leaves us with 3. And since factored form is best, we'll write our second fraction as 5 times x plus 2, still with that denominator x minus 1. And on the right hand side, 0 times, well, who cares, it's going to be 0. Now we still have this rational expression here, but if we multiply everything by x minus 1, we'll eliminate the next fraction. And so we'll multiply every term by x minus 1. And so the first term, that becomes 3 times x minus 1. The second term, the common factor of x minus 1 will be removed, leaving us with 5 times x plus 2. And on the right hand side, we have 0 times something, which is going to give us 0. And now we have a much simpler equation to solve. So let's expand, collect our like terms, and solve. And at this point, it's worth remembering that while we have a solution, a forbidden value can never be a solution. And so we need to make sure that this solution is not one of the forbidden values. And since the only forbidden values are negative 2 and 1, x equals negative 7 and 8 is fine, and so that will be our solution. So again, anytime you have a rational expression, the very first and most important thing you do is identify what is forbidden. And so we find that x cannot be 0 and x cannot be negative 2. And now we can get rid of the fractions. If we multiply through by the common denominator, x times x plus 2, we can eliminate all of the fractions. So we'll multiply our first fraction by x times x plus 2, our second fraction by x times x plus 2, and the right hand side also times x times x plus 2. We'll remove the common factors. In the first product, x is a common factor so that vanishes and we're left with 5 times x plus 2. In the second product, x plus 2 is a common factor, so we'll remove it and we'll have 4x left. And over on the right hand side, x times x plus 2 is a common factor, so that vanishes and leaving us with 1. And now we'll solve the equation. And since x equals negative 1 is not one of the forbidden values, it is the solution. It's important to find the forbidden values before you do anything else. For example, we have this denominator of x plus 2, so we'll eliminate it by multiplying everything by x plus 2. We have a denominator of x minus 1, so we'll multiply everything by x minus 1. And now we have a nice equation, so we'll solve it. And we find that this is the wrong answer. So why is this answer wrong? Remember that a forbidden value can never be a solution. And if we take a look at our original equation, we realize that we have to require that x cannot be 1 and x cannot be negative 2. And what that means is that this thing that looks like a solution is actually one of the forbidden values. And since this is a forbidden value, this is not a solution. And in fact, this equation has no solution.