Simplify (1/(x^2)-1/(3x))/(9/x-x)

$\frac{\frac{1}{x^{2}} - \frac{1}{3 x}}{\frac{9}{x} - x}$

Multiply the numerator and denominator of the complex fraction by $3 x^{2}$ .

Multiply $\frac{\frac{1}{x^{2}} - \frac{1}{3 x}}{\frac{9}{x} - x}$ by $\frac{3 x^{2}}{3 x^{2}}$ .

$\frac{3 x^{2}}{3 x^{2}} \cdot \frac{\frac{1}{x^{2}} - \frac{1}{3 x}}{\frac{9}{x} - x}$

Combine.

$\frac{3 x^{2} (\frac{1}{x^{2}} - \frac{1}{3 x})}{3 x^{2} (\frac{9}{x} - x)}$

Apply the distributive property.

$\frac{3 x^{2} \frac{1}{x^{2}} + 3 x^{2} (- \frac{1}{3 x})}{3 x^{2} \frac{9}{x} + 3 x^{2} (- x)}$

Simplify by cancelling.

Cancel the common factor of $x^{2}$ .

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{x^{2} \cdot 3 \frac{1}{x^{2}} + 3 x^{2} (- \frac{1}{3 x})}{3 x^{2} \frac{9}{x} + 3 x^{2} (- x)}$

Cancel the common factor.

$\frac{x^{2} \cdot 3 \frac{1}{x^{2}} + 3 x^{2} (- \frac{1}{3 x})}{3 x^{2} \frac{9}{x} + 3 x^{2} (- x)}$

Rewrite the expression.

$\frac{3 + 3 x^{2} (- \frac{1}{3 x})}{3 x^{2} \frac{9}{x} + 3 x^{2} (- x)}$

Cancel the common factor of $3 x$ .

Move the leading negative in $- \frac{1}{3 x}$ into the numerator.

$\frac{3 + 3 x^{2} \frac{- 1}{3 x}}{3 x^{2} \frac{9}{x} + 3 x^{2} (- x)}$

Factor $3 x$ out of $3 x^{2}$ .

$\frac{3 + 3 x (x) \frac{- 1}{3 x}}{3 x^{2} \frac{9}{x} + 3 x^{2} (- x)}$

Cancel the common factor.

$\frac{3 + 3 x \cdot x \frac{- 1}{3 x}}{3 x^{2} \frac{9}{x} + 3 x^{2} (- x)}$

Rewrite the expression.

$\frac{3 + x \cdot - 1}{3 x^{2} \frac{9}{x} + 3 x^{2} (- x)}$

Cancel the common factor of $x$ .

Factor $x$ out of $3 x^{2}$ .

$\frac{3 + x \cdot - 1}{x (3 x) \frac{9}{x} + 3 x^{2} (- x)}$

Cancel the common factor.

$\frac{3 + x \cdot - 1}{x (3 x) \frac{9}{x} + 3 x^{2} (- x)}$

Rewrite the expression.

$\frac{3 + x \cdot - 1}{3 x \cdot 9 + 3 x^{2} (- x)}$

Multiply $9$ by $3$ .

$\frac{3 + x \cdot - 1}{27 x + 3 x^{2} (- x)}$

Simplify the numerator.

Move $- 1$ to the left of $x$ .

$\frac{3 - 1 \cdot x}{27 x + 3 x^{2} (- x)}$

Rewrite $- 1 x$ as $- x$ .

$\frac{3 - x}{27 x + 3 x^{2} (- x)}$

Simplify the denominator.

Factor $3 x$ out of $27 x + 3 x^{2} (- x)$ .

Factor $3 x$ out of $27 x$ .

$\frac{3 - x}{3 x (9) + 3 x^{2} (- x)}$

Factor $3 x$ out of $3 x^{2} (- x)$ .

$\frac{3 - x}{3 x (9) + 3 x (x (- x))}$

Factor $3 x$ out of $3 x (9) + 3 x (x (- x))$ .

$\frac{3 - x}{3 x (9 + x (- x))}$

Combine exponents.

Raise $x$ to the power of $1$ .

$\frac{3 - x}{3 x (9 - (x^{1} x))}$

Raise $x$ to the power of $1$ .

$\frac{3 - x}{3 x (9 - (x^{1} x^{1}))}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 - x}{3 x (9 - x^{1 + 1})}$

Add $1$ and $1$ .

$\frac{3 - x}{3 x (9 - x^{2})}$

Rewrite $9$ as $3^{2}$ .

$\frac{3 - x}{3 x (3^{2} - x^{2})}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

$\frac{3 - x}{3 x (3 + x) (3 - x)}$

Cancel the common factor of $3 - x$ .

Cancel the common factor.

$\frac{3 - x}{3 x (3 + x) (3 - x)}$

Rewrite the expression.

$\frac{1}{3 x (3 + x)}$

Simplify (1/(x^2)-1/(3x))/(9/x-x)

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