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The inequality you have provided is:

\[
\frac{x+4}{x-3} \leq 0
\]

### Steps to solve:

1. **Find Critical Points**:
   - The numerator \(x + 4 = 0\) gives \(x = -4\).
   - The denominator \(x - 3 = 0\) gives \(x = 3\), but \(x = 3\) causes a vertical asymptote or division by zero, so it cannot be part of the solution.

2. **Sign Analysis**:
   - To determine where the expression \(\frac{x+4}{x-3}\) is negative or zero, we need to evaluate the signs of the expression in intervals divided by the critical points \(x = -4\) and \(x = 3\).
   - The intervals to test are: \((- \infty, -4)\), \((-4, 3)\), and \((3, \infty)\).

3. **Test the intervals**:
   - For \(x \in (-\infty, -4)\), let's test \(x = -5\): 
     \[
     \frac{-5+4}{-5-3} = \frac{-1}{-8} = \frac{1}{8} > 0
     \]
   - For \(x \in (-4, 3)\), let's test \(x = 0\): 
     \[
     \frac{0+4}{0-3} = \frac{4}{-3} < 0
     \]
   - For \(x \in (3, \infty)\), let's test \(x = 4\): 
     \[
     \frac{4+4}{4-3} = \frac{8}{1} > 0
     \]

4. **Include Zero**:
   - At \(x = -4\), the expression equals zero: 
     \[
     \frac{-4+4}{-4-3} = \frac{0}{-7} = 0
     \]
     So \(x = -4\) is part of the solution.
   - At \(x = 3\), the expression is undefined, so it is not part of the solution.

### Final Solution:
The inequality holds in the interval \([-4, 3)\).

Thus, the solution is:
\[
x \in [-4, 3)
\]

Would you like more details or have any questions about this solution?

### Related Questions:
1. How do we handle inequalities with more complex rational expressions?
2. What happens if the inequality sign is \(>\) or \(<\)?
3. How do we test intervals when dealing with rational inequalities?
4. What other methods can we use to solve rational inequalities?
5. Can this approach be extended to higher-degree polynomials?

### Tip:
Always remember to exclude values that make the denominator zero when solving rational inequalities!

Solve the inequality \( \frac{x+4}{x-3} \leq 0 \).

This guide explains how to solve the rational inequality \( \frac{x+4}{x-3} \leq 0 \) using interval testing and critical points analysis. Learn to find the intervals where the inequality holds and understand the concept of asymptotes. Suitable for students in Grades 9-11.

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