College Algebra
Tutorial 15: Equations with Rational Expressions
Learning Objectives
After completing this tutorial, you should be able to:
 Solve equations with rational expressions.
 Know if a solution is an extraneous solution or not.

Introduction
The equations that we will be working with in this section all have
rational expressions ( fractions  yuck!). After a few magical
steps, we can transform these equations with rational expressions into
linear equations. From there, you solve the linear equation like
you normally do. If you need a review on how to solve a linear equation,
feel free to go to Tutorial 14:
Linear Equations in One Variable.
No matter what type of equation you are working with in this section,
the ultimate goal is to get your variable on one side and everything else
on the other side using inverse operations. 
Tutorial
Solving Rational Equations

Step 1: Simplify
by removing the fractions.
We do this by multiplying both sides by the LCD.
If you need a review on finding the LCD of a rational expression, go
to Tutorial 10: Adding and Subtracting
Rational Expressions.
Note that even though this is not the same as adding and subtracting
rational expressions, you still find the LCD in the same manner. So
if you go to this link, just look at finding the LCD, NOT adding and subtracting
rational expressions. 
Step 2: Solve the
remaining equation.
In this tutorial the remaining equations will all be linear.
If you need a review on solving linear equations go back to Tutorial
14: Linear Equations. 
Step 3: Check for
extraneous solutions.
For rational equations, extraneous solutions are values that cause
any denominator in the original problem to be 0. Of course, when
we have 0 in the denominator we have an expression that is undefined.
So, we would have to throw out any values that would cause the denominator
to be 0. 
In Tutorial
14: Linear Equations, I told you that when you multiply both sides
by the same constant that the two sides would remain equal to each other.
But we can not guarantee that if you are multiplying by an expression that
has the variable you are solving for  which is the situation we will be
running into in this section. Sometimes this will cause extraneous
solutions. 
Step 1: Simplify
by removing the fractions. 

*Inverse of add. 24 is sub. 24
*Inverse of mult. by 1 is div. by 1

Note that 9 does not cause any denominators to be zero. So it
is not an extraneous solution. 9 is the solution to our equation. 
Step 1: Simplify
by removing the fractions. 

*Remove ( ) by using dist. prop.
*Get all a terms
on one side
*Inverse of add. 8 is sub. 8

Note that 7 does not cause any denominators to be zero. So it
is not an extraneous solution. 7 is the solution to our equation. 
Step 1: Simplify
by removing the fractions. 

*Remove ( ) by using dist. prop.
*Inverse of add. 9 is sub. 9
*Inverse of mult. by 1 is div. by 1

Note that 3 does cause two of the denominators to be zero.
So 3 is an extraneous solution. That means there is no
solution.
The answer is NO solution. 
Practice Problems
These are practice problems to help bring you to the next level.
It will allow you to check and see if you have an understanding of these
types of problems. Math works just like anything
else, if you want to get good at it, then you need to practice it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice. To get the most out of these, you should work the problem out on
your own and then check your answer by clicking on the link for the answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer. 
Practice
Problems 1a  1b: Solve the given equation.
Need Extra Help on these Topics?
WTAMU > Virtual Math Lab > College Algebra
Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.
Last revised on Dec 16, 2009 by Kim Seward.
All contents copyright (C) 2002  2010, WTAMU and Kim Seward.
All rights reserved.

