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We are asked to solve the rational inequality:

\[
\frac{x - 4}{x + 8} < 0
\]

### Steps to solve:

1. **Find the critical points**: These occur where the numerator and denominator are equal to zero.

   - **Numerator**: \(x - 4 = 0 \implies x = 4\)
   - **Denominator**: \(x + 8 = 0 \implies x = -8\)

2. **Divide the number line**: Use the critical points to divide the number line into intervals:
   - Interval 1: \( (-\infty, -8) \)
   - Interval 2: \( (-8, 4) \)
   - Interval 3: \( (4, \infty) \)

3. **Test each interval**: Pick a test point from each interval and substitute it into the expression \( \frac{x - 4}{x + 8} \) to determine the sign.

   - For \( x = -9 \) (Interval 1: \( (-\infty, -8) \)):
     \[
     \frac{-9 - 4}{-9 + 8} = \frac{-13}{-1} = 13 > 0
     \]
     So, the expression is positive in this interval.

   - For \( x = 0 \) (Interval 2: \( (-8, 4) \)):
     \[
     \frac{0 - 4}{0 + 8} = \frac{-4}{8} = -\frac{1}{2} < 0
     \]
     So, the expression is negative in this interval.

   - For \( x = 5 \) (Interval 3: \( (4, \infty) \)):
     \[
     \frac{5 - 4}{5 + 8} = \frac{1}{13} > 0
     \]
     So, the expression is positive in this interval.

4. **Determine the solution**: We need the expression to be less than zero, so we take the interval where it is negative.

   The expression is negative in \( (-8, 4) \).

5. **Handle the endpoints**: 
   - \( x = -8 \) makes the denominator zero, so it is undefined, and we exclude it.
   - \( x = 4 \) makes the numerator zero, but we are looking for strictly less than zero, so exclude it.

### Final Solution:

The solution set is:

\[
(-8, 4)
\]

Would you like further details on any part of this process?

Here are some related questions:

1. How do you solve rational inequalities in general?
2. What happens when the inequality is greater than or equal to 0 instead?
3. How do you handle more complex rational expressions in inequalities?
4. Why do we exclude points where the denominator is zero?
5. Can a rational inequality have more than two critical points?

**Tip**: Always remember to check for undefined points (where the denominator is zero) in rational inequalities, as they can affect the solution interval.

Solve the rational inequality: (x - 4) / (x + 8) < 0. Express the solution set in interval notation.

Learn how to solve the rational inequality (x - 4) / (x + 8) < 0 with a step-by-step explanation, including number line analysis, critical points, and interval testing. Suitable for high school students in Grades 9-11.

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