# Chapter R - Elementary Algebra Review - R.6 Rational Expressions and Equations - R.6 Exercise Set - Page 980: 36

The simplified form of a complex rational expression $\frac{\frac{x}{3}-\frac{3}{x}}{\frac{1}{x}+\frac{1}{3}}$ is $x-3$.

#### Work Step by Step

$\frac{\frac{x}{3}-\frac{3}{x}}{\frac{1}{x}+\frac{1}{3}}$ The denominators within the complex rational expression are $x\text{ and }3$. Thus, the LCD is $3x$. Multiply the expression with $\frac{3x}{3x}$: $\frac{\frac{x}{3}-\frac{3}{x}}{\frac{1}{x}+\frac{1}{3}}=\frac{\frac{x}{3}-\frac{3}{x}}{\frac{1}{x}+\frac{1}{3}}\cdot \frac{3x}{3x}$ Apply the distributive property, \begin{align} & \frac{\frac{x}{3}-\frac{3}{x}}{\frac{1}{x}+\frac{1}{3}}=\frac{\frac{x}{3}\cdot 3x-\frac{3}{x}\cdot 3x}{\frac{1}{x}\cdot 3x+\frac{1}{3}\cdot 3x} \\ & =\frac{x\cdot x-3\cdot 3}{1\cdot 3+1\cdot x} \end{align} Simplify the terms, \begin{align} & \frac{\frac{x}{3}-\frac{3}{x}}{\frac{1}{x}+\frac{1}{3}}=\frac{{{x}^{2}}-9}{3+x} \\ & =\frac{{{x}^{2}}-9}{x+3} \end{align} \begin{align} & \frac{\frac{x}{3}-\frac{3}{x}}{\frac{1}{x}+\frac{1}{3}}=\frac{\left( x+3 \right)\left( x-3 \right)}{x+3} \\ & =x-3 \end{align}

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