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.