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!