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The inequality provided is:

\[
\frac{x - 1}{x + 4} \leq 0
\]

### Steps to solve the inequality:

1. **Identify critical points:**
   - The numerator \( x - 1 = 0 \) gives a critical point \( x = 1 \).
   - The denominator \( x + 4 = 0 \) gives a critical point \( x = -4 \) (but note that the expression is undefined at \( x = -4 \)).

2. **Break the real line into intervals based on the critical points:**
   The critical points split the real number line into three intervals:
   - \( (-\infty, -4) \)
   - \( (-4, 1) \)
   - \( (1, \infty) \)

3. **Test each interval:**
   For each interval, substitute a value into the inequality to check whether the expression is positive or negative.

   - For \( (-\infty, -4) \), take \( x = -5 \):
     \[
     \frac{-5 - 1}{-5 + 4} = \frac{-6}{-1} = 6 > 0
     \]
     So, the expression is **positive** in this interval.
     
   - For \( (-4, 1) \), take \( x = 0 \):
     \[
     \frac{0 - 1}{0 + 4} = \frac{-1}{4} = -\frac{1}{4} < 0
     \]
     So, the expression is **negative** in this interval.
     
   - For \( (1, \infty) \), take \( x = 2 \):
     \[
     \frac{2 - 1}{2 + 4} = \frac{1}{6} > 0
     \]
     So, the expression is **positive** in this interval.

4. **Include the boundary points:**
   - At \( x = 1 \), the numerator becomes zero, so \( \frac{1 - 1}{1 + 4} = 0 \), which satisfies the inequality.
   - At \( x = -4 \), the expression is undefined, so \( x = -4 \) is excluded from the solution.

5. **Conclusion:**
   The solution to the inequality is the interval where the expression is less than or equal to zero, which is:
   \[
   x \in (-4, 1]
   \]

Would you like more details or have any questions?

### Related Questions:
1. How would the solution change if the inequality was \( \frac{x - 1}{x + 4} \geq 0 \)?
2. What happens if we change the numerator to \( x + 1 \) in the inequality?
3. How do you solve a rational inequality that has a strict inequality \( < \) or \( > \)?
4. Can you explain how to graph the solution \( (-4, 1] \) on the number line?
5. What is the significance of critical points in rational inequalities?

### Tip:
Always check if the denominator equals zero, since that makes the expression undefined and splits the intervals for testing the inequality.

Solve the inequality: (x - 1) / (x + 4) <= 0

Learn how to solve the rational inequality (x - 1) / (x + 4) <= 0 with a detailed step-by-step solution. This problem covers key algebraic concepts, including critical points, interval testing, and rational expressions. Suitable for students in Grades 9-12.

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