To solve the inequality:

$\frac{x - 3}{x + 6} < 1$

Step 1: Move all terms to one side.

We can rewrite the inequality by subtracting 1 from both sides:

$\frac{x - 3}{x + 6} - 1 < 0$

Step 2: Combine the terms.

To combine the terms into a single fraction, we need a common denominator:

$\frac{x - 3}{x + 6} - \frac{x + 6}{x + 6} < 0$

Simplify the numerator:

$\frac{(x - 3) - (x + 6)}{x + 6} < 0$

Simplifying further:

$\frac{x - 3 - x - 6}{x + 6} < 0$

$\frac{-9}{x + 6} < 0$

Step 3: Analyze the inequality.

Now we need to solve the inequality:

$\frac{-9}{x + 6} < 0$

For a fraction to be less than 0, the numerator and denominator must have opposite signs. Since the numerator $-9$ is always negative, we require the denominator $(x + 6)$ to be positive:

$x + 6 > 0 \quad \Rightarrow \quad x > -6$

Step 4: Conclusion.

Thus, the solution to the inequality is:

$x > -6$

Final Solution:

$x > -6$

Would you like further clarification or details? Here are some related questions to consider:

1. How does solving rational inequalities differ from solving linear inequalities?
2. What happens if the inequality sign were reversed?
3. Can you solve similar inequalities with different numerators?
4. How do you handle inequalities with multiple terms in the denominator?
5. What would the solution be if the inequality was $\frac{x-3}{x+6} \leq 1$?

Tip: Always check for points where the denominator could be zero, as these are not in the domain of the inequality.