Simplify 5-(x+2)÷((x^2-4)/(x-3))

$5 - (x + 2) \div (\frac{x^{2} - 4}{x - 3})$

Step 1

Simplify each term.

To divide by a fraction, multiply by its reciprocal.

$5 - ((x + 2) \frac{x - 3}{x^{2} - 4})$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

$5 - \frac{x - 3}{x - 2}$

Step 2

To write $5$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Apply the distributive property.

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

Multiply $5$ by $- 2$ .

$\frac{5 x - 10 - (x - 3)}{x - 2}$

Apply the distributive property.

$\frac{5 x - 10 - x - - 3}{x - 2}$

Multiply $- 1$ by $- 3$ .

$\frac{5 x - 10 - x + 3}{x - 2}$

Subtract $x$ from $5 x$ .

$\frac{4 x - 10 + 3}{x - 2}$

Add $- 10$ and $3$ .

$\frac{4 x - 7}{x - 2}$