Simplify (3/x-x/3)/((x+2)/3-1/x)

$\frac{\frac{3}{x} - \frac{x}{3}}{\frac{x + 2}{3} - \frac{1}{x}}$

Multiply the numerator and denominator of the complex fraction by $x \cdot 3$ .

Multiply $\frac{\frac{3}{x} - \frac{x}{3}}{\frac{x + 2}{3} - \frac{1}{x}}$ by $\frac{x \cdot 3}{x \cdot 3}$ .

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

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

Cancel the common factor of $x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $3$ .

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

Cancel the common factor of $3$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $- 1$ .

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

Simplify the numerator.

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

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

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 $b = x$ .

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

Simplify the denominator.

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

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

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

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 $2$ .

$- 1, 3$

Write the factored form using these integers.

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

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (3/x-x/3)/((x+2)/3-1/x)

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