https://file.aiquickdraw.com/mathbot/file/1726148722245wecyfcla.jpg

The equation in the image is:

\[
\frac{x - 4}{x + 5} \leq 0
\]

### Steps to solve:

1. **Find critical points**: The critical points occur when the numerator or the denominator is zero.
   - \( x - 4 = 0 \) gives \( x = 4 \) (numerator zero).
   - \( x + 5 = 0 \) gives \( x = -5 \) (denominator zero).

2. **Sign analysis**: To determine where the expression is less than or equal to zero, we will perform sign analysis using the intervals created by the critical points \( x = -5 \) and \( x = 4 \). These intervals are:
   - \( (-\infty, -5) \)
   - \( (-5, 4) \)
   - \( (4, \infty) \)

3. **Test the sign in each interval**:
   - For \( (-\infty, -5) \), pick \( x = -6 \):
     \[
     \frac{-6 - 4}{-6 + 5} = \frac{-10}{-1} = 10 \quad (\text{positive})
     \]
   - For \( (-5, 4) \), pick \( x = 0 \):
     \[
     \frac{0 - 4}{0 + 5} = \frac{-4}{5} \quad (\text{negative})
     \]
   - For \( (4, \infty) \), pick \( x = 5 \):
     \[
     \frac{5 - 4}{5 + 5} = \frac{1}{10} \quad (\text{positive})
     \]

4. **Include or exclude critical points**:
   - At \( x = 4 \), the numerator is zero, so \( \frac{0}{x + 5} = 0 \), which satisfies the \( \leq 0 \) condition.
   - At \( x = -5 \), the denominator is zero, which makes the expression undefined, so exclude \( x = -5 \).

5. **Solution**: The expression is less than or equal to zero in the interval:
   \[
   (-5, 4]
   \]

### Final Answer:
\[
x \in (-5, 4]
\]

Would you like further details or have any questions?

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Here are 5 related questions:
1. How do you perform sign analysis for rational inequalities?
2. What happens when the denominator is zero in a rational inequality?
3. Can a rational inequality have multiple solutions?
4. How do we graph the solution of \( \frac{x - 4}{x + 5} \leq 0 \)?
5. How do the numerator and denominator affect the sign of a rational expression?

**Tip**: Always check whether critical points should be included in the solution set, especially where the denominator is zero!

Solve the rational inequality \( \frac{x - 4}{x + 5} \leq 0 \).

This guide explains how to solve the rational inequality \( \frac{x - 4}{x + 5} \leq 0 \) using sign analysis and critical points. Follow the step-by-step solution, including finding the critical points and determining the intervals where the inequality holds. Suitable for students in Grades 9-11.

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