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It looks like you are working with a rational inequality. Based on the uploaded image, the inequality appears to be structured as follows:

\[
\frac{2x - 12}{3x + 6} \geq 0
\]

### Let's break down the steps to solve this rational inequality:

1. **Factor both the numerator and the denominator:**

   - Numerator: \(2x - 12 = 2(x - 6)\)
   - Denominator: \(3x + 6 = 3(x + 2)\)

   Thus, the inequality becomes:

   \[
   \frac{2(x - 6)}{3(x + 2)} \geq 0
   \]

2. **Find the critical points:**
   
   The critical points are where the numerator or denominator equals zero.

   - Numerator: \(x - 6 = 0 \implies x = 6\)
   - Denominator: \(x + 2 = 0 \implies x = -2\)
   
   These points split the number line into intervals.

3. **Test the intervals:**

   We now test points in the intervals created by the critical points: \((-∞, -2)\), \((-2, 6)\), and \((6, ∞)\).

   - For \(x = -3\) (in \((-∞, -2)\)): 
     \[
     \frac{2(-3 - 6)}{3(-3 + 2)} = \frac{2(-9)}{3(-1)} = \frac{-18}{-3} = 6 > 0
     \]
   
   - For \(x = 0\) (in \((-2, 6)\)): 
     \[
     \frac{2(0 - 6)}{3(0 + 2)} = \frac{2(-6)}{3(2)} = \frac{-12}{6} = -2 < 0
     \]
   
   - For \(x = 7\) (in \((6, ∞)\)): 
     \[
     \frac{2(7 - 6)}{3(7 + 2)} = \frac{2(1)}{3(9)} = \frac{2}{27} > 0
     \]

4. **Determine the sign changes:**

   The expression is positive in the intervals \((-∞, -2)\) and \((6, ∞)\), and negative in \((-2, 6)\).

5. **Include the endpoints:**

   - At \(x = -2\), the expression is undefined (division by zero).
   - At \(x = 6\), the expression is zero, so \(x = 6\) satisfies the inequality.

### Final Solution:

\[
x \in (-∞, -2) \cup [6, ∞)
\]

Would you like further details on any of these steps or a deeper explanation?

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Here are 5 related questions that might help:

1. How do we handle rational inequalities where the inequality is strictly greater or strictly less than zero?
2. What happens if the critical points include complex numbers?
3. Can you explain how rational expressions behave near their asymptotes?
4. How do you graph the solution to a rational inequality?
5. How do critical points affect the intervals of the solution?

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

Solve the rational inequality: (2x - 12) / (3x + 6) >= 0.

Step-by-step solution of the rational inequality (2x - 12) / (3x + 6) >= 0. This problem involves factoring, determining critical points, and analyzing sign changes. Suitable for students in Grades 9-12.

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