To solve the inequality:

\[
\frac{x + 3}{3x - 6} > 0
\]

### Step 1: Factor the denominator
We can factor the denominator \( 3x - 6 \) as follows:
\[
3x - 6 = 3(x - 2)
\]
Thus, the inequality becomes:
\[
\frac{x + 3}{3(x - 2)} > 0
\]

### Step 2: Determine critical points
The critical points occur when either the numerator or the denominator is equal to zero. These points will help us identify intervals to test the sign of the expression.

- Numerator: \( x + 3 = 0 \) gives \( x = -3 \)
- Denominator: \( x - 2 = 0 \) gives \( x = 2 \)

So the critical points are \( x = -3 \) and \( x = 2 \).

### Step 3: Analyze the sign of the expression in each interval
We now divide the real line into intervals based on the critical points: 
\[
(-\infty, -3), (-3, 2), (2, \infty)
\]
For each interval, we will test a value to determine if the expression is positive or negative.

1. **Interval \( (-\infty, -3) \)**: Test with \( x = -4 \)
   \[
   \frac{-4 + 3}{3(-4 - 2)} = \frac{-1}{3(-6)} = \frac{-1}{-18} = \frac{1}{18} > 0
   \]
   So, the expression is **positive** in this interval.

2. **Interval \( (-3, 2) \)**: Test with \( x = 0 \)
   \[
   \frac{0 + 3}{3(0 - 2)} = \frac{3}{3(-2)} = \frac{3}{-6} = -\frac{1}{2} < 0
   \]
   So, the expression is **negative** in this interval.

3. **Interval \( (2, \infty) \)**: Test with \( x = 3 \)
   \[
   \frac{3 + 3}{3(3 - 2)} = \frac{6}{3(1)} = \frac{6}{3} = 2 > 0
   \]
   So, the expression is **positive** in this interval.

### Step 4: Check the boundary points
- At \( x = -3 \), the numerator is zero, so the expression is \( 0 \), which does not satisfy the inequality.
- At \( x = 2 \), the denominator is zero, and division by zero is undefined.

### Step 5: Conclusion
The solution to the inequality \( \frac{x + 3}{3(x - 2)} > 0 \) is:

\[
x \in (-\infty, -3) \cup (2, \infty)
\]

Would you like any further details or have any questions?

---

Here are some related questions to explore further:
1. What happens if we have a different inequality, like \( \frac{x + 3}{3x - 6} < 0 \)?
2. How would the solution change if the inequality was \( \frac{x + 3}{3(x - 2)} \geq 0 \)?
3. Can you solve a similar inequality with different factors, like \( \frac{x^2 + 3}{x - 1} > 0 \)?
4. How do you solve an equation involving absolute values, such as \( |x + 3| > 5 \)?
5. How do you handle quadratic inequalities, like \( x^2 - 5x + 6 > 0 \)?

**Tip:** When solving inequalities, always check the boundary points separately, as they might lead to undefined values or zero solutions!

Master the step-by-step process of solving the rational inequality (x + 3) / (3(x - 2)) > 0. Learn how to factor, find critical points, and analyze the sign of the expression across intervals. Perfect for students in Grades 9-11 looking to understand rational inequalities.

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