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.