Answer
$$x = \frac{4}{3}$$
Work Step by Step
First we need to determine the LCM of the equation. Using prime factorization: $$x+3: x+3$$ $$x-3: x-3$$ $$x^{2}-9: x^{2}-3^{2} = (x+3)(x-3)$$
The expression comprised of factors that appear in at least one of the factored expressions is the LCM: $$(x+3)(x-3)$$
Multiply the whole equation by the LCM: $$(\frac{2}{x+3})(x+3)(x-3) = (\frac{1}{x^{2}-9})(x+3)(x-3) - (\frac{1}{x-3})(x+3)(x-3)$$
Simplify and solve for $x$: $$2(x-3) = 1 - (x+3)$$ $$2x - 6 = 1 - x - 3$$ $$2x + x = 1 - 3 + 6$$ $$3x = 4$$
Dividing both sides by $3$: $$x = \frac{4}{3}$$
