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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.

Solve the absolute value inequality: |(3x + 9)/3| < 6. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

Learn how to solve the absolute value inequality |(3x + 9)/3| < 6 with step-by-step instructions. This solution simplifies the expression inside the absolute value and breaks down the compound inequality into a clear interval notation solution: (-9, 3). Suitable for students in Grades 9-10.

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