To solve the inequality \(|6 + 3x| < 9\), we need to consider the definition of absolute value. Recall that for any expression \( |A| < B \), this implies \(-B < A < B\).

### Step-by-Step Solution:

1. **Set up the compound inequality:**

   Given \( |6 + 3x| < 9 \), we can write:
   \[
   -9 < 6 + 3x < 9
   \]

2. **Solve the inequality in two parts:**

   Let's solve the two inequalities separately:

   - For the left inequality: \(-9 < 6 + 3x\)
   - For the right inequality: \(6 + 3x < 9\)

   **Solve \(-9 < 6 + 3x\):**

   \[
   -9 - 6 < 3x
   \]

   \[
   -15 < 3x
   \]

   Divide both sides by 3:

   \[
   -5 < x
   \]

   **Solve \(6 + 3x < 9\):**

   \[
   3x < 9 - 6
   \]

   \[
   3x < 3
   \]

   Divide both sides by 3:

   \[
   x < 1
   \]

3. **Combine the results:**

   From both parts, we have:

   \[
   -5 < x < 1
   \]

   This represents the solution to the inequality.

### Final Answer:

\[
\boxed{-5 < x < 1}
\]

This means \(x\) can take any value between \(-5\) and \(1\), not including \(-5\) or \(1\).

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Would you like more details on this solution, or do you have any other questions?

### Related Questions:
1. How do you solve absolute value inequalities with greater than signs, such as \(|x - 4| > 7\)?
2. What are the steps to solve a quadratic inequality?
3. How do you solve compound inequalities with different operations?
4. Can you explain how to graph the solution set of \(-5 < x < 1\) on a number line?
5. What changes when the absolute value inequality has a "greater than or equal to" sign?

**Tip:** Always check your solutions by plugging the boundary values back into the original inequality to ensure they satisfy the condition.

Learn how to solve the inequality |6 + 3x| < 9 with a detailed step-by-step solution. Understand the concept of absolute value and inequalities. Suitable for high school students studying algebra and inequalities.

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