We are tasked with solving the inequality:

\[
2|x + 9| + 1 > 7
\]

### Step 1: Simplify the inequality
Start by subtracting 1 from both sides to isolate the absolute value expression:

\[
2|x + 9| > 6
\]

### Step 2: Divide by 2
Next, divide both sides by 2 to simplify further:

\[
|x + 9| > 3
\]

### Step 3: Solve the absolute value inequality
To remove the absolute value, break it into two cases:

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

#### Case 1: \( x + 9 > 3 \)
Subtract 9 from both sides:

\[
x > -6
\]

#### Case 2: \( x + 9 < -3 \)
Subtract 9 from both sides:

\[
x < -12
\]

### Step 4: Combine the solutions
The solution to this inequality is:

\[
x > -6 \quad \text{or} \quad x < -12
\]

In interval notation, the solution is:

\[
(-\infty, -12) \cup (-6, \infty)
\]

Would you like further clarification on any steps? 

Here are 5 related questions:
1. How do you solve absolute value equations?
2. What are the differences between solving absolute value inequalities and equations?
3. Can you explain interval notation in detail?
4. How do you graph solutions of inequalities on a number line?
5. What are the properties of absolute value functions?

**Tip:** When solving absolute value inequalities, always split them into two cases to remove the absolute value!

Learn how to solve the absolute value inequality 2|x + 9| + 1 > 7 step by step. This solution covers essential algebra concepts, including breaking down absolute value inequalities into two cases and solving them. Ideal for students in Grades 9-10.

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