(x+5)/(x) <= 0
Solve the rational inequality. Write your answer in interval notation.

x+5x≤0

To solve the inequality \(\frac{x+5}{x} \leq 0\), follow these steps:

1. **Find critical points**:
   - The numerator \(x + 5 = 0\) gives \(x = -5\).
   - The denominator \(x = 0\) must also be considered, as it cannot be included in the solution.

2. **Test intervals**:
   The critical points divide the number line into the following intervals: 
   - \( (-\infty, -5) \)
   - \( (-5, 0) \)
   - \( (0, \infty) \)

3. **Choose test points**:
   - For \(x = -6\) in \((- \infty, -5)\):
     \[
     \frac{-6 + 5}{-6} = \frac{-1}{-6} > 0
     \]
   - For \(x = -1\) in \((-5, 0)\):
     \[
     \frac{-1 + 5}{-1} = \frac{4}{-1} < 0
     \]
   - For \(x = 1\) in \((0, \infty)\):
     \[
     \frac{1 + 5}{1} = \frac{6}{1} > 0
     \]

4. **Determine where the inequality holds**:
   - The inequality \(\frac{x+5}{x} \leq 0\) is satisfied in the interval \((-5, 0)\).

5. **Include endpoints**:
   - Since \(x = -5\) makes the expression zero, it is included in the solution.
   - \(x = 0\) cannot be included because it makes the expression undefined.

Thus, the solution in interval notation is:

\[
[-5, 0)
\]

### Summary of the solution:
- The inequality \(\frac{x+5}{x} \leq 0\) holds for \(x \in [-5, 0)\).

If you have any questions or need further details, let me know! Here are some related questions to consider:

1. What happens to the inequality if we change the direction of the inequality sign?
2. How would the solution change if the inequality was strict (i.e., \(<\) instead of \(\leq\))?
3. Can you explain how to find the signs of rational expressions?
4. What is the graphical representation of this inequality?
5. How do you solve a rational inequality with more complex expressions in the numerator and denominator? 

### Tip:
Always check the signs of the expression in each interval by using test points to ensure you correctly identify where the inequality holds true.

Discover how to solve the rational inequality (x+5)/x <= 0 with a comprehensive step-by-step guide. This problem explores critical points, testing intervals, and interval notation, making it ideal for high school students in Grades 9-12.

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