Simplify ((x+5)/6+1/x)/(1/(x^2)-1/9)

$\frac{\frac{x + 5}{6} + \frac{1}{x}}{\frac{1}{x^{2}} - \frac{1}{9}}$

Multiply the numerator and denominator of the complex fraction by $18 x^{2}$ .

Multiply $\frac{\frac{x + 5}{6} + \frac{1}{x}}{\frac{1}{x^{2}} - \frac{1}{9}}$ by $\frac{18 x^{2}}{18 x^{2}}$ .

$\frac{18 x^{2}}{18 x^{2}} \cdot \frac{\frac{x + 5}{6} + \frac{1}{x}}{\frac{1}{x^{2}} - \frac{1}{9}}$

Combine.

$\frac{18 x^{2} (\frac{x + 5}{6} + \frac{1}{x})}{18 x^{2} (\frac{1}{x^{2}} - \frac{1}{9})}$

Apply the distributive property.

$\frac{18 x^{2} \frac{x + 5}{6} + 18 x^{2} \frac{1}{x}}{18 x^{2} \frac{1}{x^{2}} + 18 x^{2} (- \frac{1}{9})}$

Simplify by cancelling.

Cancel the common factor of $6$ .

Factor $6$ out of $18 x^{2}$ .

$\frac{6 (3 x^{2}) \frac{x + 5}{6} + 18 x^{2} \frac{1}{x}}{18 x^{2} \frac{1}{x^{2}} + 18 x^{2} (- \frac{1}{9})}$

Cancel the common factor.

$\frac{6 (3 x^{2}) \frac{x + 5}{6} + 18 x^{2} \frac{1}{x}}{18 x^{2} \frac{1}{x^{2}} + 18 x^{2} (- \frac{1}{9})}$

Rewrite the expression.

$\frac{3 x^{2} (x + 5) + 18 x^{2} \frac{1}{x}}{18 x^{2} \frac{1}{x^{2}} + 18 x^{2} (- \frac{1}{9})}$

Cancel the common factor of $x$ .

Factor $x$ out of $18 x^{2}$ .

$\frac{3 x^{2} (x + 5) + x (18 x) \frac{1}{x}}{18 x^{2} \frac{1}{x^{2}} + 18 x^{2} (- \frac{1}{9})}$

Cancel the common factor.

$\frac{3 x^{2} (x + 5) + x (18 x) \frac{1}{x}}{18 x^{2} \frac{1}{x^{2}} + 18 x^{2} (- \frac{1}{9})}$

Rewrite the expression.

$\frac{3 x^{2} (x + 5) + 18 x}{18 x^{2} \frac{1}{x^{2}} + 18 x^{2} (- \frac{1}{9})}$

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

Factor $x^{2}$ out of $18 x^{2}$ .

$\frac{3 x^{2} (x + 5) + 18 x}{x^{2} \cdot 18 \frac{1}{x^{2}} + 18 x^{2} (- \frac{1}{9})}$

Cancel the common factor.

$\frac{3 x^{2} (x + 5) + 18 x}{x^{2} \cdot 18 \frac{1}{x^{2}} + 18 x^{2} (- \frac{1}{9})}$

Rewrite the expression.

$\frac{3 x^{2} (x + 5) + 18 x}{18 + 18 x^{2} (- \frac{1}{9})}$

Cancel the common factor of $9$ .

Move the leading negative in $- \frac{1}{9}$ into the numerator.

$\frac{3 x^{2} (x + 5) + 18 x}{18 + 18 x^{2} \frac{- 1}{9}}$

Factor $9$ out of $18 x^{2}$ .

$\frac{3 x^{2} (x + 5) + 18 x}{18 + 9 (2 x^{2}) \frac{- 1}{9}}$

Cancel the common factor.

$\frac{3 x^{2} (x + 5) + 18 x}{18 + 9 (2 x^{2}) \frac{- 1}{9}}$

Rewrite the expression.

$\frac{3 x^{2} (x + 5) + 18 x}{18 + 2 x^{2} \cdot - 1}$

Multiply $- 1$ by $2$ .

$\frac{3 x^{2} (x + 5) + 18 x}{18 - 2 x^{2}}$

Simplify the numerator.

Factor $3 x$ out of $3 x^{2} (x + 5) + 18 x$ .

Factor $3 x$ out of $3 x^{2} (x + 5)$ .

$\frac{3 x (x (x + 5)) + 18 x}{18 - 2 x^{2}}$

Factor $3 x$ out of $18 x$ .

$\frac{3 x (x (x + 5)) + 3 x (6)}{18 - 2 x^{2}}$

Factor $3 x$ out of $3 x (x (x + 5)) + 3 x (6)$ .

$\frac{3 x (x (x + 5) + 6)}{18 - 2 x^{2}}$

Apply the distributive property.

$\frac{3 x (x \cdot x + x \cdot 5 + 6)}{18 - 2 x^{2}}$

Multiply $x$ by $x$ .

$\frac{3 x (x^{2} + x \cdot 5 + 6)}{18 - 2 x^{2}}$

Move $5$ to the left of $x$ .

$\frac{3 x (x^{2} + 5 \cdot x + 6)}{18 - 2 x^{2}}$

Factor $x^{2} + 5 x + 6$ 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 $6$ and whose sum is $5$ .

$2, 3$

Write the factored form using these integers.

$\frac{3 x ((x + 2) (x + 3))}{18 - 2 x^{2}}$

$\frac{3 x (x + 2) (x + 3)}{18 - 2 x^{2}}$

Simplify the denominator.

Factor $2$ out of $18 - 2 x^{2}$ .

Factor $2$ out of $18$ .

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

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

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

Factor $2$ out of $2 (9) + 2 (- x^{2})$ .

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

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

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

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 $b = x$ .

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

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify ((x+5)/6+1/x)/(1/(x^2)-1/9)

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