Solving Equations Containing
Rational Expressions
Example 1
Solve:
Solution
Step 1 Find the LCD.

LCD = 2 Â· 3 Â· x = 6x 
Step 2 Multiply by the LCD. 

Step 3 Distribute on the right side.


Step 4 Reduce. 

Step 5 Solve.


Simplify.
Subtract 27 from both sides.
Multiply both sides by 1. 
18 = 27  x 9 = x
9 = x 
Step 6 Check the solution.
We leave the check to you.


Thus, the solution is x = 9. 

Note:
Hereâ€™s one way to find the LCD of
â€¢ Factor each
denominator:
x = x
2x = 2 Â· x
6 = 2 Â· 3
â€¢ List each factor the
greatest number of
times it appears in
any factorization. x, 2, 3
â€¢ Multiply. 2 Â· 3
Â· x = 6x
So, the LCD is 6x.
Example 2
Solve:
Solution
Step 1 Find the LCD.

LCD = (x  5)(x + 5) 
Step 2 Multiply by the LCD. 

Step 3 Distribute the LCD to both terms
the right side. 

Step 4 Reduce. 

Step 5 Solve.


Simplify.
Distribute on the right side.
Combine like terms.
Add 5 to both sides.
Divide both sides by 3.

16
16
16
21
7 
= x + 5 + (x  5) Â· 2
= x + 5 + 2x  10
= 3x  5
= 3x
= x 
Step 6 Check the solution.
We leave the check to you.


Thus, the solution is x = 7.
