To simplify the expression
[tex]\[
\frac{\frac{3}{x^2-9}}{\frac{3}{x+3} + \frac{4}{x-3}}
\][/tex]
we start by simplifying both the numerator and the denominator.
### Step 1: Factor the denominator of the numerator
The numerator is [tex]\(\frac{3}{x^2 - 9}\)[/tex]. Notice that [tex]\(x^2 - 9\)[/tex] can be factored as a difference of squares:
[tex]\[
x^2 - 9 = (x + 3)(x - 3)
\][/tex]
So, we can rewrite the numerator as:
[tex]\[
\frac{3}{(x + 3)(x - 3)}
\][/tex]
### Step 2: Simplify the entire denominator
The denominator is [tex]\(\frac{3}{x + 3} + \frac{4}{x - 3}\)[/tex]. To combine these fractions, we need a common denominator. The common denominator is [tex]\((x + 3)(x - 3)\)[/tex]. Rewrite the fractions with this common denominator:
[tex]\[
\frac{3}{x + 3} = \frac{3(x - 3)}{(x + 3)(x - 3)} = \frac{3x - 9}{(x + 3)(x - 3)}
\][/tex]
[tex]\[
\frac{4}{x - 3} = \frac{4(x + 3)}{(x + 3)(x - 3)} = \frac{4x + 12}{(x + 3)(x - 3)}
\][/tex]
Now add these two fractions:
[tex]\[
\frac{3x - 9 + 4x + 12}{(x + 3)(x - 3)} = \frac{7x + 3}{(x + 3)(x - 3)}
\][/tex]
### Step 3: Put it all together
Now we can rewrite our original complex fraction:
[tex]\[
\frac{\frac{3}{(x + 3)(x - 3)}}{\frac{7x + 3}{(x + 3)(x - 3)}}
\][/tex]
Since we are dividing by a fraction, we multiply by the reciprocal:
[tex]\[
\frac{3}{(x + 3)(x - 3)} \times \frac{(x + 3)(x - 3)}{7x + 3}
\][/tex]
The [tex]\((x + 3)(x - 3)\)[/tex] terms cancel out:
[tex]\[
\frac{3}{7x + 3}
\][/tex]
So the simplified expression is:
[tex]\[
\frac{3}{7x + 3}
\][/tex]
### Final Answer
Thus, the correct answer is:
[tex]\[
(A) \frac{3}{7x + 3}
\][/tex]
