The given expression is:

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

Step-by-Step Solution

1. Factor the denominator $x^2 - 9$: The expression $x^2 - 9$ is a difference of squares, so it can be factored as: $x^2 - 9 = (x - 3)(x + 3)$

   So the new expression is: $\frac{3}{x - 3} - \frac{x}{(x - 3)(x + 3)}$

2. Find a common denominator: The least common denominator (LCD) between $x - 3$ and $(x - 3)(x + 3)$ is $(x - 3)(x + 3)$.

3. Rewrite both terms with the common denominator: The first term $\frac{3}{x - 3}$ can be rewritten as: $\frac{3}{x - 3} = \frac{3(x + 3)}{(x - 3)(x + 3)}$

   Now the expression becomes: $\frac{3(x + 3)}{(x - 3)(x + 3)} - \frac{x}{(x - 3)(x + 3)}$

4. Combine the terms: Since both terms now have the same denominator, we can combine the numerators: $\frac{3(x + 3) - x}{(x - 3)(x + 3)}$

5. Simplify the numerator: Expand and simplify the numerator: $3(x + 3) = 3x + 9$ So the numerator becomes: $(3x + 9 - x) = 2x + 9$

   Now the expression is: $\frac{2x + 9}{(x - 3)(x + 3)}$

Final Answer:

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

Would you like any further details or have questions?

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Related Questions:

1. How do you factor a difference of squares in general?
2. What are the steps to find the least common denominator in a rational expression?
3. How do you simplify complex rational expressions?
4. Can you explain how to handle undefined points in a rational function?
5. What is the domain of the simplified expression?

Tip: Always check for common factors between numerators and denominators before combining terms to simplify expressions.