 A rational equation is an equation involving a rational expression. We can solve them in two ways. One way is based on something that we might call the zero quotient property. Suppose I have a quotient x over y equal to zero, and we require that y can't be zero. In that case, we know that x must be zero. So this means we can solve our equation involving rational expressions by rearranging so the equation is equal to zero, combining the rational expressions into a single rational expression equal to zero, requiring the denominator not be zero, and requiring the numerator to be equal to zero. For example, let's try and solve this rational expression. So first, we'll rearrange so the equation is equal to zero, done. We'll combine the rational expressions to a single rational expression equal to zero, done. Now we have to require the denominator not be equal to zero. And so since the denominator can't be zero, we'll see what would make it equal to zero. So we'll have our denominator equal to zero, and we'll solve. And because we're fantastically lucky, this one happens to be factorable. Now I have product equal to zero, and so one or the other factor must be zero. So either x plus five is zero, or x minus one is zero. Solving these gives us. And since these are the values that make the denominator equal to zero, we require that x can't be negative five, and x can't be one. Now our rational expression is essentially a fraction, and since this is a fraction equal to zero, we know the numerator must be zero. And so we must have three x minus seven equal to zero, and we can solve this. And again, the most important thing to remember is that we do have some forbidden values here. And so we can never let x be negative five or one. But since x equals seven thirds is not a forbidden value, it must be our solution. Well, let's take a slightly more complicated expression. So again, anything that would make a denominator equal to zero must be forbidden. So we require that x not be negative two, and x can't be equal to one. Now this is a sum of rational expressions. So in order to make use of our property, we need a single rational expression equal to zero. And so that means we'll have to add these rational expressions. Now if you had a kind and gentle math teacher in a kind and gentle universe, you'd be given rational expressions with the same denominator. But you're not that lucky, you have me. And that means you have rational expressions that do not have the same denominator. And so we have to find a common denominator. And the thing to remember is that a common denominator of x over y and c over w is the product of the two denominators. And so a common denominator will be, and we'll convert each rational expression into a rational expression with the denominator x plus two times x minus one. This first rational expression is missing the x minus one factor. So we'll multiply numerator and denominator by x minus one to get. And likewise this second fraction is missing the factor x plus two. So we'll multiply numerator and denominator to get. And remember that factored form is best. So unless we have a good reason to multiply them out, we'll leave things in factored form. Now since our rational expressions have the same denominators, we can add the numerators. And unfortunately at this point we do need to expand and multiply to simplify our numerator. So let's take our numerator and expand and simplify, which gives us our equation. And so now we have rational expression equal to zero. And so for a rational expression, if x over y is equal to zero and y is not equal to zero, then our numerator must be zero. So we'll set our numerator equal to zero and solve. And we should check that this is not one of the forbidden values, and it isn't. And so x equals minus seven-eighths will be the solution.