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The image shows a set of rational inequalities along with corresponding solution sets labeled A through J. The task is likely to match each inequality to its correct solution. Let's work through them step by step.

### Process:
For each rational inequality, we must analyze the sign of the rational expression and find where it holds true, focusing on critical points like the zeros of the numerator and the denominator.

Let's solve each inequality and match it to one of the solutions:

---

1. **Inequality:**  
   \[
   \frac{x+7}{x+4} > 0
   \]  
   **Critical points:**  
   - Numerator zero at \( x = -7 \)  
   - Denominator zero at \( x = -4 \) (undefined)  
   **Sign analysis:**  
   - For \( x < -7 \), both the numerator and denominator are negative, so \( \frac{x+7}{x+4} > 0 \).  
   - For \( -7 < x < -4 \), the numerator is positive and the denominator is negative, so \( \frac{x+7}{x+4} < 0 \) (false).  
   - For \( x > -4 \), both the numerator and denominator are positive, so \( \frac{x+7}{x+4} > 0 \).  
   **Solution:**  
   \( (-\infty, -7) \cup (-4, \infty) \)  
   Matches **B**: \( x < -6 \text{ or } x > 9 \)

---

2. **Inequality:**  
   \[
   \frac{x+5}{x-6} \leq 0
   \]  
   **Critical points:**  
   - Numerator zero at \( x = -5 \)  
   - Denominator zero at \( x = 6 \) (undefined)  
   **Sign analysis:**  
   - For \( x < -5 \), both the numerator and denominator are negative, so \( \frac{x+5}{x-6} > 0 \).  
   - For \( -5 < x < 6 \), numerator is positive, denominator negative, so \( \frac{x+5}{x-6} < 0 \).  
   **Solution:**  
   \( (-5, 6) \)  
   Matches **A**: \( -5 \leq x <

Match the following rational inequalities with their correct solution sets.

Learn how to solve rational inequalities step-by-step by analyzing critical points and conducting sign analysis. This problem set involves finding solutions for inequalities like (x+7)/(x+4) > 0 and matching them to their correct solution sets. Suitable for high school students, Grades 10-12.

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