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.