Solving a Rational Equation
This post about solving a rational equation is #15 in a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.
Solve this rational equation:
Assume x is not equal to -8 or 2/3.
What do we mean by rational? The root is ratio. A rational number is a number that can be expressed as quotient — a ratio — of two integers. A rational expression is a quotient — a ratio — of algebraic expressions. A rational equation is an equation comprised of rational expressions.
To get started solving this rational equation, multiply both sides by . That should give you:
Cancel the terms that show up in both numerator and denominator. That’s on the left and on the right. That leaves:
(Yes, I could have just said “cross multiply,” but I hate that expression. It’s a personal quirk.) Now we’re done with denominators.
Subtract from both sides, and then switch the sides.
For more about factoring, see “How to Factor a Quadratic.”
Set each factor equal to 0:
Remember that division by zero yields something undefined, so an x-value that makes a denominator zero is not a valid answer. So you need to make sure that the x values you’ve found do not make one of the denominators in the original question zero. To find those values, set those denominators equal to zero:
Oh, look: Those are the values that the question says are excluded. The question is telling you that a solution that causes the denominator to be zero is not a valid solution. But today that is not your problem, because that’s not one of the answers we found. We found and . Interesting factoid: The official practice question on which this question is based stipulates that x is not equal to -3/2 and then offers -3/2 as an answer choice. If there’s a lesson in that, it may be that it’s a good idea to reread the question before you give your final answer.
To complete the check, substitute and into the original equation.
Related: Rationalizing a Denominator and How Do You Simplify a Rational Expression within a Rational Expression?
This question is similar to question number 15 in the sample questions for the Accuplacer Advanced Algebra and Functions test.