Extraneous Solutions
If an equation contains a variable in the denominator of a rational
expression, then it is possible that the equation may not have a solution.
For example, letâ€™s try
to solve this equation:



Multiply both sides
by (x  4). 


Cancel. 


Distribute the 2.
Simplify the right side.
Add 12 to both sides. 
x  12 x  12
x  12
x 
= (x  4) Â· 2  2x
= 2x  8  2x
= 8
= 4 
The result is x = 4.
However, notice what happens when we check the solution. 

Substitute 4 for x:
Simplify. 


The result is division by zero, which is undefined.
This means that x = 4 is not a solution.
Therefore, the equation has no solution.
We call 4 an extraneous solution. It is a number that results from
solving an equation, but it is not a solution of the equation.
Since it does not satisfy the original equation, an extraneous solution
is not really a solution. An extraneous solution is sometimes called a
false solution.
Example
Solve:
Solution
Multiply each side by x(x + 3), the LCD of the rational expression.
Distribute x(x + 3) to each term on the right side.
Cancel common factors.

4x 
= x + 3  12 
Subtract x from both sides
and simplify the right side. 
3x 
= 9 
Divide both sides by 3. 
x 
= 3 
It appears that x = 3 is a solution.



However, if we substitute 3 for x in the original equation
the result is division by 0. 

Division by 0 is undefined, so x = 3 is not a solution.
Therefore, the equation has no solution.
