We are asked to simplify the complex rational expression:

$\frac{1 - \frac{5}{x}}{1 + \frac{5}{x}}$

Step 1: Simplify the numerator and denominator

The numerator is $1 - \frac{5}{x}$, and the denominator is $1 + \frac{5}{x}$. To simplify each, we rewrite them as:

$\text{Numerator: } 1 - \frac{5}{x} = \frac{x}{x} - \frac{5}{x} = \frac{x - 5}{x}$

$\text{Denominator: } 1 + \frac{5}{x} = \frac{x}{x} + \frac{5}{x} = \frac{x + 5}{x}$

Step 2: Rewrite the entire expression

Now the original complex fraction becomes:

$\frac{\frac{x - 5}{x}}{\frac{x + 5}{x}}$

Step 3: Simplify the complex fraction

We simplify by dividing the two fractions. When dividing fractions, multiply by the reciprocal of the denominator:

$\frac{x - 5}{x} \div \frac{x + 5}{x} = \frac{x - 5}{x} \times \frac{x}{x + 5} = \frac{x - 5}{x + 5}$

Final Answer:

The simplified expression is:

$\boxed{\frac{x - 5}{x + 5}}$

So the correct answer is A. $\frac{x - 5}{x + 5}$.

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Would you like further clarification or a more detailed explanation? Here are five related questions to deepen your understanding:

1. How do you handle complex fractions with different denominators?
2. What is the importance of multiplying by the reciprocal in dividing fractions?
3. How do you simplify expressions with mixed numbers and fractions?
4. Can you simplify $\frac{1 - \frac{3}{y}}{1 + \frac{3}{y}}$ using a similar approach?
5. What happens when you simplify a complex fraction with higher-degree polynomials?

Tip: Always factor numerators and denominators when possible before simplifying complex fractions.