Checkpoint: Complex Fractions Solution

Simplify.

$\frac{\frac{6}{x^{2} + 7 x - 18} - \frac{1}{x - 2}}{\frac{1}{x + 9} - 1}$

$Numerator \begin{matrix} Denominator \end{matrix} \frac{6}{(x - 2) (x + 9)} - \frac{1 (x + 9)}{(x - 2) (x + 9)} \begin{matrix} \frac{1}{x + 9} - \frac{x + 9}{x + 9} \end{matrix} 6 - x - 9 = - x - 3 \begin{matrix} 1 - x - 9 = - x - 8 \end{matrix} \frac{- (x + 3)}{(x - 2) (x + 9)}, x \neq 2, - 9 \begin{matrix} \frac{- (x + 8)}{x + 9}, x \neq - 9 \end{matrix}$

$\frac{- (x + 3)}{(x - 2) (x + 9)} \div \frac{- (x + 8)}{x + 9} \frac{- (x + 3)}{(x - 2) (x + 9)} \cdot \frac{x + 9}{- (x + 8)} \frac{- (x + 3) (x + 9)}{- (x - 2) (x + 9) (x + 8)} \frac{x + 3}{(x - 2) (x + 8)}, x \neq - 9, - 8, 2$

Note

Q: What should be done before writing the expression horizontally?
A: Simplify the numerator and denominator by finding the LCD and combining the numerator and denominator into one expression each.

Remember that –1 can be factored out of an expression, especially when all terms are negative.

Pay careful attention to the signs of the terms, as there is subtraction in the numerator and denominator of the complex fraction.