We need to solve the inequality \( |3x + 9| > 7 \). Here's how we can approach it step-by-step:

### Step 1: Set up two cases for the absolute value.

For any absolute value inequality \( |A| > B \), where \( B > 0 \), the inequality splits into two cases:
1. \( A > B \)
2. \( A < -B \)

In this case, \( A = 3x + 9 \) and \( B = 7 \). Therefore, we get the following two inequalities:

1. \( 3x + 9 > 7 \)
2. \( 3x + 9 < -7 \)

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### Step 2: Solve each inequality.

#### Case 1: \( 3x + 9 > 7 \)
\[
3x > 7 - 9
\]
\[
3x > -2
\]
\[
x > -\frac{2}{3}
\]

#### Case 2: \( 3x + 9 < -7 \)
\[
3x < -7 - 9
\]
\[
3x < -16
\]
\[
x < -\frac{16}{3}
\]

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### Step 3: Combine the solutions.

The solution to \( |3x + 9| > 7 \) is the union of the two intervals:

\[
x < -\frac{16}{3} \quad \text{or} \quad x > -\frac{2}{3}
\]

In interval notation, the solution is:

\[
(-\infty, -\frac{16}{3}) \cup (-\frac{2}{3}, \infty)
\]

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Would you like further details or have any questions?  

Here are 5 related questions:  
1. How do you solve absolute value inequalities involving \( \leq \) or \( \geq \)?  
2. How does the solution change if the inequality was \( |3x + 9| < 7 \)?  
3. What is the general rule for splitting absolute value inequalities?  
4. How can you graph the solution on a number line?  
5. What happens if the inequality is \( |A| > -B \) with \( B < 0 \)?  

**Tip:** Always check if the absolute value expression can be simplified before solving the inequality.

Learn to solve the absolute value inequality |3x + 9| > 7 with a step-by-step solution. This tutorial explores how to break down and solve absolute value inequalities, covering concepts in algebra suitable for Grades 8-10.

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