https://file.aiquickdraw.com/mathbot/file/1726183442655ixh468ce.jpg

The inequality shown in the image is:

\[
\frac{x - 6}{(x + 3)(x + 5)} \geq 0
\]

### Steps to solve the inequality:

1. **Find the critical points**:
   The critical points occur when the numerator is zero or when the denominator is zero.

   - **Numerator**: \( x - 6 = 0 \) gives \( x = 6 \).
   - **Denominator**: 
     - \( x + 3 = 0 \) gives \( x = -3 \),
     - \( x + 5 = 0 \) gives \( x = -5 \).

   So, the critical points are \( x = -5, -3, 6 \).

2. **Determine the sign of the expression**:
   To determine where the expression is positive or negative, check the sign of the expression in the intervals formed by the critical points: \( (-\infty, -5) \), \( (-5, -3) \), \( (-3, 6) \), and \( (6, \infty) \).

   - For \( x < -5 \), the numerator \( x - 6 \) is negative, and both factors of the denominator are negative, so the expression is positive.
   - For \( -5 < x < -3 \), the numerator is negative, \( x + 3 \) is negative, and \( x + 5 \) is positive, so the expression is negative.
   - For \( -3 < x < 6 \), the numerator is negative, and both factors of the denominator are positive, so the expression is negative.
   - For \( x > 6 \), the numerator is positive, and both factors of the denominator are positive, so the expression is positive.

3. **Check the critical points**:
   - At \( x = -5 \) and \( x = -3 \), the denominator becomes zero, so the expression is undefined.
   - At \( x = 6 \), the numerator is zero, so the expression equals zero.

4. **Conclusion**:
   The solution to the inequality is where the expression is greater than or equal to zero. Therefore, the solution is:
   
   \[
   (-\infty, -5) \cup [6, \infty)
   \]

### Would you like further details or clarifications on this solution?

Here are five related questions to explore:
1. What is the significance of the critical points in inequalities?
2. How do you test signs in different intervals of a rational inequality?
3. How would the solution change if the inequality were strict (\( > 0 \))?
4. What does it mean for a function to be undefined at certain points?
5. How do rational inequalities apply in real-world scenarios?

**Tip**: Always check the behavior of the inequality at critical points and exclude points where the denominator is zero.

Solve the inequality (x - 6) / [(x + 3)(x + 5)] ≥ 0.

Learn how to solve the rational inequality (x - 6) / [(x + 3)(x + 5)] ≥ 0. This solution covers key mathematical concepts, including identifying critical points and analyzing signs across intervals. Suitable for high school students in Grades 10-12.

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