 Since one of the important goals of algebra is to be able to solve all types of equations, then every time we get introduced to a new type of expression, we have a new type of equation. So remember that a rational expression is a quotient of two polynomials, like this, this, or this. Hey, wait a minute. That's not a rational expression. Get out of here! Now, the important thing is that since we can't divide by zero, every rational expression could have forbidden values. A rational equation is an equation involving a rational expression. And whenever you have rational expressions, the important thing to remember is to always identify values that make the expression undefined. So here's a nice rational equation. Let's begin by identifying any forbidden values. Since no denominator can be zero, anything that makes a denominator zero is forbidden. So let's see what we have to avoid by setting each denominator equal to zero. So if x plus two equals zero, then x is... If x plus four is zero, then x is... In order for the product x plus two times x plus four to be zero, one of them must be zero. So either x plus two is zero, well, we've already solved that. Or x plus four is zero, and we've already solved that. So it's important to keep track of the logic. x equals negative two, x equals negative four make a denominator zero, and that is forbidden. And so we have to require x not be negative two, and x not be negative four. So we can simplify rational equations by using two useful ideas. First, we can multiply all terms of an equation by the same thing and still have an equation. And second, when we multiply a fraction by its denominator, we eliminate the denominator. And what this means is that we can use this approach to rewrite any rational equation as a polynomial equation which we can solve. However, the new equation may have solutions equal to the forbidden values. And the thing to remember is that a forbidden value is never a solution. So if our new equation does have some solutions equal to the forbidden values, these are extraneous solutions and should be omitted. So let's start off with this rational equation. And the first thing we do is identify any forbidden values, and so we require x not be equal to negative two or negative four. The first denominator is x plus two, so if we multiply all terms by x plus two, and now, because all of our fractions are product over product, we can begin cancellation. And since we've done something, we've obviously solved the problem. Or maybe not. Don't be a politician. Always ask, what else can we do? And we still have a denominator of x plus four, so if we multiply all of our terms by x plus four. And now we have a series of fractions, so we can cancel out the common factors. And now we have an equation where there are no fractions. So now we'll solve the equation. Over on the left we have a mess of arithmetic operations, so we'll expand. And after the dust settles, we see that we have a quadratic equation. So we'll pick our favorite method of solving quadratic equations. How about the quadratic formula? And we get solutions of x equals three and x equals negative four. And we can circle these as our answers, but remember that a forbidden value is never a solution. And we've required that x was not equal to negative two and not equal to negative four. And so that means this solution, x equals negative four, is extraneous and should not be included in our final answer. Or let's take a look at this equation. So first we'll find the forbidden values, those that make the denominator equal to zero. That would be x equal to negative three and x equal to two. We'll multiply through by our denominators x plus three and then x minus two. Now we have a whole mess of arithmetic, so we'll expand everything out and collect like terms. And after all the dust settles, we end up with a quadratic equation. And let's go straight to using the quadratic formula. And again, we have two solutions to the new equation. We have to make sure that these aren't any of the forbidden values. And so here the easy thing to recognize is that both of these are irrational numbers. So neither of them can be equal to two or negative three. So they are both solutions.