Solving Rational Equations
After studying this lesson, you will be able to:
 Solve rational equations.
To solve an equation with fractions (rational equations),
multiply each term by a common denominator. This will eliminate
the fractions. Then, solve the equation that remains.
Example 1
The common denominator is 6x so we multiply each term by 6x:

Now, we have 3 multiplications to do 
this will eliminate the variables 
Collect like terms 
29 = x 

**remember, the solution cannot cause any denominator to equal
zero  if it does, it is an extraneous solution
Example 2
The common denominator is 60 so we multiply each term by 60:

Now, we have 3 multiplications to do 
this will eliminate the variables 
30  15r + 36r = 20r + 40 
Collect like terms 
30 + 21r = 20r + 40 
Get the variables together by subtracting
20r from each side 
30 + r = 40 
Subtract 30 from each side 
r = 10 

Example 3
To find the common denominator we have to factor the second
denominator:
The common denominator is (m + 1) (m  1) so we multiply each
term by (m + 1) (m  1):
Now, we have 3 multiplications to do  cancel out identical
binomials
2m ( m + 1 ) + (m + 3 ) = m^{ 2}
 1 
Multiply 
2m^{ 2}  2m + m + 3 = m^{ 2}
 1 
Collect like terms. 
2m^{ 2}  m + 3 = m^{ 2}
 1 
This is a quadratic equation so we need
to set it equal to zero. Let's move all terms to the
right side. 
0 = 3m^{ 2} + m  4 

Now, factor and set each factor equal to zero and solve.
0 = (3m + 4) ( m  1 )
3m + 4 = 0 
m  1 = 0 

m = 1 
m = 1 is an extraneous solution because it will cause a
denominator to equal zero.
