Simplify Complex Rational Expressions: example 8.56

Simplify: $\frac{\frac{2}{x + 6}}{\frac{4}{x - 6} - \frac{4}{x^{2} - 36}} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Simplify: $\frac{\frac{3}{x + 2}}{\frac{5}{x - 2} - \frac{3}{x^{2} - 4}} .$

How did it go?


Find the LCD of all fractions in the complex rational expression. The LCD is $(x + 6) (x - 6)$ .
Multiply the numerator and denominator by the LCD.
Simplify the expression.
Distribute in the denominator.
Simplify.
Simplify.
To simplify the denominator, distribute and combine like terms.
Remove common factors.
Simplify.
Notice that there are no more factors common to the numerator and denominator.


Find the LCD of all fractions in the complex rational expression. The LCD is $(x + 6) (x - 6)$ .
Multiply the numerator and denominator by the LCD.
Simplify the expression.
Distribute in the denominator.
Simplify.
Simplify.
To simplify the denominator, distribute and combine like terms.
Remove common factors.
Simplify.
Notice that there are no more factors common to the numerator and denominator.