Simplify the complex fraction.
-$\frac { \frac { 9 } { x } - \frac { x } { 9 } } { \frac { 1 } { 9 } - \frac { 1 } { x } }$
A) $x + 9$
B) $9 x ( x + 9 )$
C) $- ( x - 9 )$
D) $- ( x + 9 )$

Question

Quizplus · Accepted Answer

To simplify the given complex fraction, we first find a common denominator for the numerators and denominators within the complex fraction. The common denominator for terms involving x and 9 is 9x. We rewrite each fraction with this common denominator:Numerator: $\frac{9}{x} - \frac{x}{9} = \frac{81 - x^2}{9x}$Denominator: $\frac{1}{9} - \frac{1}{x} = \frac{x - 9}{9x}$The complex fraction becomes $\frac{\frac{81 - x^2}{9x}}{\frac{x - 9}{9x}}$. Simplifying this, we divide the numerators and cancel the denominators, which gives us $\frac{81 - x^2}{x - 9}$.Factoring the numerator as a difference of squares, we get $\frac{(9 - x)(9 + x)}{x - 9}$.Notice that $(9 - x)$ and $(x - 9)$ are negatives of each other, so they cancel out to -1, leaving us with $- (x + 9)$, which corresponds to choice D.

Simplify the Complex Fraction 9x−x919−1x\frac { \frac { 9 } { x } - \frac { x } { 9 } } { \frac { 1 } { 9 } - \frac { 1 } { x } }91​−x1​x9​−9x​​

Simplify the Complex Fraction $\frac { \frac { 9 } { x } - \frac { x } { 9 } } { \frac { 1 } { 9 } - \frac { 1 } { x } }$