Simplify 3/(x-3)+2/(x+3)-(4x)/(x^2-9)

$\frac{3}{x - 3} + \frac{2}{x + 3} - \frac{4 x}{x^{2} - 9}$

Step 1

Simplify the denominator.

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

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

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 $b = 3$ .

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

Step 2

To write $\frac{3}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

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

Step 3

To write $\frac{2}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

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

Step 4

Write each expression with a common denominator of $(x - 3) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $(x - 3) (x + 3)$ .

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

Step 5

Simplify terms.

Combine the numerators over the common denominator.

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

Combine the numerators over the common denominator.

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

Step 6

Simplify each term.

Apply the distributive property.

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

Multiply $3$ by $3$ .

$\frac{3 x + 9 + 2 (x - 3) - 4 x}{(x + 3) (x - 3)}$

Apply the distributive property.

$\frac{3 x + 9 + 2 x + 2 \cdot - 3 - 4 x}{(x + 3) (x - 3)}$

Multiply $2$ by $- 3$ .

$\frac{3 x + 9 + 2 x - 6 - 4 x}{(x + 3) (x - 3)}$

Step 7

Simplify terms.

Add $3 x$ and $2 x$ .

$\frac{5 x + 9 - 6 - 4 x}{(x + 3) (x - 3)}$

Subtract $4 x$ from $5 x$ .

$\frac{x + 9 - 6}{(x + 3) (x - 3)}$

Subtract $6$ from $9$ .

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

Cancel the common factor of $x + 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify 3/(x-3)+2/(x+3)-(4x)/(x^2-9)

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