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

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

Step 1

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

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

Factor $x$ out of $x \cdot 9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

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

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

Factor $3$ out of $x \cdot 9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $3$ .

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

Cancel the common factor of $3$ .

Factor $3$ out of $x \cdot 9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $9$ .

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

$\frac{9 + x \cdot - 3}{x \cdot 3 + x \cdot 9 \frac{- x}{9}}$

Factor $9$ out of $x \cdot 9$ .

$\frac{9 + x \cdot - 3}{x \cdot 3 + 9 \cdot x \frac{- x}{9}}$

Cancel the common factor.

$\frac{9 + x \cdot - 3}{x \cdot 3 + 9 \cdot x \frac{- x}{9}}$

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Step 4

Simplify the numerator.

Factor $3$ out of $9 + x \cdot - 3$ .

Factor $3$ out of $9$ .

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

Factor $3$ out of $x \cdot - 3$ .

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

Factor $3$ out of $3 (3) + 3 (x \cdot - 1)$ .

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

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

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

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

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

Step 5

Simplify terms.

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

Factor $x$ out of $x \cdot 3$ .

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

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

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

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

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

Cancel the common factor of $3 - x$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{3}{x}$

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

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