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

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

Step 1

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

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x + 3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $2$ .

$\frac{6 + (x + 3) \cdot 3 (- \frac{1}{3})}{(x + 3) \cdot 3 \frac{x^{2}}{3} + (x + 3) \cdot 3 \cdot - 3}$

Cancel the common factor of $3$ .

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

$\frac{6 + (x + 3) \cdot 3 \frac{- 1}{3}}{(x + 3) \cdot 3 \frac{x^{2}}{3} + (x + 3) \cdot 3 \cdot - 3}$

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

$\frac{6 + 3 \cdot (x + 3) \frac{- 1}{3}}{(x + 3) \cdot 3 \frac{x^{2}}{3} + (x + 3) \cdot 3 \cdot - 3}$

Cancel the common factor.

$\frac{6 + 3 \cdot (x + 3) \frac{- 1}{3}}{(x + 3) \cdot 3 \frac{x^{2}}{3} + (x + 3) \cdot 3 \cdot - 3}$

Rewrite the expression.

$\frac{6 + (x + 3) \cdot - 1}{(x + 3) \cdot 3 \frac{x^{2}}{3} + (x + 3) \cdot 3 \cdot - 3}$

Cancel the common factor of $3$ .

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

$\frac{6 + (x + 3) \cdot - 1}{3 \cdot (x + 3) \frac{x^{2}}{3} + (x + 3) \cdot 3 \cdot - 3}$

Cancel the common factor.

$\frac{6 + (x + 3) \cdot - 1}{3 \cdot (x + 3) \frac{x^{2}}{3} + (x + 3) \cdot 3 \cdot - 3}$

Rewrite the expression.

$\frac{6 + (x + 3) \cdot - 1}{(x + 3) x^{2} + (x + 3) \cdot 3 \cdot - 3}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{6 + x \cdot - 1 + 3 \cdot - 1}{(x + 3) x^{2} + (x + 3) \cdot 3 \cdot - 3}$

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

$\frac{6 - 1 \cdot x + 3 \cdot - 1}{(x + 3) x^{2} + (x + 3) \cdot 3 \cdot - 3}$

Multiply $3$ by $- 1$ .

$\frac{6 - 1 \cdot x - 3}{(x + 3) x^{2} + (x + 3) \cdot 3 \cdot - 3}$

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

$\frac{6 - x - 3}{(x + 3) x^{2} + (x + 3) \cdot 3 \cdot - 3}$

Subtract $3$ from $6$ .

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

Step 5

Simplify the denominator.

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

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

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

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

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

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

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

Multiply $3$ by $- 3$ .

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

Rewrite $x^{2} - 9$ in a factored form.

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

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

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

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

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

Combine exponents.

Raise $x + 3$ to the power of $1$ .

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

Raise $x + 3$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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

Factor $- 1$ out of $- x$ .

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

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

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

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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