We are given the absolute value inequality:

$\left| \frac{3x + 9}{3} \right| < 6$

Step 1: Simplify the expression inside the absolute value

First, simplify the expression inside the absolute value:

$\frac{3x + 9}{3} = x + 3$

So, the inequality becomes:

$|x + 3| < 6$

Step 2: Solve the absolute value inequality

To solve $|x + 3| < 6$, we rewrite it as a compound inequality:

$-6 < x + 3 < 6$

Step 3: Solve for $x$

Now, subtract 3 from all parts of the inequality:

$-6 - 3 < x < 6 - 3$ $-9 < x < 3$

Step 4: Write the solution in interval notation

The solution in interval notation is:

$(-9, 3)$

Thus, the correct choice is A, and the solution set in interval notation is $(-9, 3)$.

Do you want more details, or do you have any other questions?

Here are five related questions to further your understanding:

1. How would the solution change if the inequality was $\geq 6$ instead of $< 6$?
2. What happens if the absolute value inequality includes $\leq 6$?
3. Can you solve $|2x - 5| < 7$ using the same approach?
4. What if the inequality were $|x + 3| > 6$? How would the solution look?
5. How do we approach inequalities that have absolute values with more complex expressions inside?

Tip: When dealing with absolute value inequalities, always convert the inequality into a compound inequality by considering both the positive and negative cases.