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!