We are tasked with simplifying the following complex fraction:

$\frac{\frac{3}{x} - 1}{\frac{9}{x^2} - 1}$

Given that $x \neq -3, 0, 3$, let's simplify step by step.

Step 1: Simplify the numerator

The numerator is:

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

We'll rewrite 1 as $\frac{x}{x}$ to have a common denominator:

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

Step 2: Simplify the denominator

The denominator is:

$\frac{9}{x^2} - 1$

We'll rewrite 1 as $\frac{x^2}{x^2}$:

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

Step 3: Substitute back into the original fraction

Now we substitute the simplified numerator and denominator:

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

Step 4: Simplify the complex fraction

To simplify the complex fraction, we multiply by the reciprocal of the denominator:

$\frac{3 - x}{x} \times \frac{x^2}{9 - x^2} = \frac{(3 - x) \cdot x^2}{x \cdot (9 - x^2)}$

Cancel out $x$ from the numerator and denominator:

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

Step 5: Factor the denominator

The denominator $9 - x^2$ is a difference of squares, so we factor it:

$9 - x^2 = (3 - x)(3 + x)$

Thus, the expression becomes:

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

Step 6: Cancel out common factors

We can cancel out $(3 - x)$ from both the numerator and the denominator:

$\frac{x}{3 + x}$

Final Answer:

The simplified expression is:

$\frac{x}{3 + x}$

Would you like further details or have any questions?

5 Related Questions:

1. How can you apply factoring to simplify rational expressions?
2. What is the significance of the difference of squares in algebra?
3. How do you handle restrictions on variables, like $x \neq 3$ or $x \neq -3$?
4. Can you explain how to find a common denominator for rational expressions?
5. How does canceling common factors simplify complex fractions?

Tip:

Always check for common factors in both the numerator and denominator before simplifying a complex fraction!