To solve the inequality \(\frac{5x + 15}{x - 6} \geq 2\), let's follow these steps:

1. **Subtract 2 from both sides:**
   \[
   \frac{5x + 15}{x - 6} - 2 \geq 0
   \]

2. **Combine into a single fraction:**
   \[
   \frac{5x + 15 - 2(x - 6)}{x - 6} \geq 0
   \]

3. **Simplify the numerator:**
   \[
   \frac{5x + 15 - 2x + 12}{x - 6} \geq 0
   \]
   \[
   \frac{3x + 27}{x - 6} \geq 0
   \]

4. **Factor the numerator:**
   \[
   \frac{3(x + 9)}{x - 6} \geq 0
   \]

5. **Determine the critical points:**
   The critical points are where the numerator or denominator is zero:
   \[
   3(x + 9) = 0 \implies x = -9
   \]
   \[
   x - 6 = 0 \implies x = 6
   \]

6. **Test intervals around the critical points:**
   Test the intervals determined by the critical points: \((-∞, -9)\), \((-9, 6)\), and \((6, ∞)\).

   - For \(x \in (-∞, -9)\):
     \[
     \frac{3(x + 9)}{x - 6} < 0 \quad (\text{since both numerator and denominator are negative, making the fraction positive})
     \]

   - For \(x \in (-9, 6)\):
     \[
     \frac{3(x + 9)}{x - 6} > 0 \quad (\text{numerator positive, denominator negative, making the fraction negative})
     \]

   - For \(x \in (6, ∞)\):
     \[
     \frac{3(x + 9)}{x - 6} > 0 \quad (\text{both numerator and denominator are positive, making the fraction positive})
     \]

7. **Combine the intervals:**
   Since we need \(\frac{3(x + 9)}{x - 6} \geq 0\), we include the critical points and positive intervals:
   \[
   x \in (-∞, -9] \cup (6, ∞)
   \]

Therefore, the solution to the inequality is:
\[
x \in (-∞, -9] \cup (6, ∞)
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you solve rational inequalities involving absolute values?
2. What is the difference between solving a strict inequality and a non-strict inequality?
3. Can you solve \(\frac{2x + 5}{x + 3} < 1\)?
4. How do you determine the sign of a rational function in different intervals?
5. Can you explain how to find vertical asymptotes in rational functions?
6. How would you solve the inequality \(x^2 - 4x - 12 < 0\)?
7. What methods can be used to solve polynomial inequalities?
8. How can you graph the solution of \(\frac{3(x + 4)}{x - 5} \geq 1\)?

### Tip:
When solving inequalities, always remember to check the critical points and consider the behavior of the function in each interval determined by these points.

Learn how to solve the rational inequality (5x + 15)/(x - 6) >= 2 with a detailed step-by-step solution. This problem involves inequalities and rational functions, focusing on finding critical points and testing intervals. Suitable for students in Grades 10-12.

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