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

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

Multiply the numerator and denominator of the complex fraction by $9 x$ .

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

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

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

Cancel the common factor of $3$ .

Factor $3$ out of $9 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

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

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

Factor $x$ out of $9 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $9$ by $- 1$ .

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

Cancel the common factor of $3$ .

Factor $3$ out of $9 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $9$ .

Move the leading negative in $- \frac{x^{2}}{9}$ into the numerator.

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

Factor $9$ out of $9 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Add $1$ and $2$ .

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

Simplify the numerator.

Factor $3$ out of $3 x (x - 2) - 9$ .

Factor $3$ out of $3 x (x - 2)$ .

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

Factor $3$ out of $- 9$ .

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

Factor $3$ out of $3 (x (x - 2)) + 3 \cdot - 3$ .

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

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Move $- 2$ to the left of $x$ .

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

Factor $x^{2} - 2 x - 3$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Write the factored form using these integers.

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

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

Simplify terms.

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

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

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

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

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

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

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

Cancel the common factor of $x - 3$ and $3 - x$ .

Factor $- 1$ out of $x$ .

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

Rewrite $- 3$ as $- 1 (3)$ .

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

Factor $- 1$ out of $- 1 (- x) - 1 (3)$ .

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the expression.

Multiply $3$ by $- 1$ .

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

Move the negative in front of the fraction.

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

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

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