Simplify ((3x)/(2x))÷((x^2-81)/(x^2+11x+18))

$\frac{3 x}{2 x} \div \frac{x^{2} - 81}{x^{2} + 11 x + 18}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x}{2 x} \cdot \frac{x^{2} + 11 x + 18}{x^{2} - 81}$

Step 2

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{3 x}{2 x} \cdot \frac{x^{2} + 11 x + 18}{x^{2} - 81}$

Rewrite the expression.

$\frac{3}{2} \cdot \frac{x^{2} + 11 x + 18}{x^{2} - 81}$

Step 3

Factor $x^{2} + 11 x + 18$ 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 $18$ and whose sum is $11$ .

$2, 9$

Write the factored form using these integers.

$\frac{3}{2} \cdot \frac{(x + 2) (x + 9)}{x^{2} - 81}$

Step 4

Simplify the denominator.

Rewrite $81$ as $9^{2}$ .

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

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 $b = 9$ .

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

Step 5

Simplify terms.

Cancel the common factor of $x + 9$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify ((3x)/(2x))÷((x^2-81)/(x^2+11x+18))

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