To solve the inequality \( 4|x + 4| + 6 > 18 \):

1. Subtract 6 from both sides:
\[
4|x + 4| + 6 - 6 > 18 - 6
\]
\[
4|x + 4| > 12
\]

2. Divide both sides by 4:
\[
|x + 4| > 3
\]

3. Now, solve the absolute value inequality. The solution will break into two cases:

- Case 1: \( x + 4 > 3 \)
\[
x > -1
\]

- Case 2: \( x + 4 < -3 \)
\[
x < -7
\]

So, the solution is:
\[
x < -7 \quad \text{or} \quad x > -1
\]

In interval notation, the solution is:
\[
(-\infty, -7) \cup (-1, \infty)
\]

Would you like more details on any part of the process?

Here are five related questions you might explore:
1. How would you solve \( 3|x - 2| + 5 = 20 \)?
2. How does solving absolute value equations differ from inequalities?
3. What happens if the inequality changes to \( 4|x + 4| + 6 \geq 18 \)?
4. How can you graph the solution of an absolute value inequality?
5. What would the solution be if the absolute value inequality was \( |x + 4| < 3 \)?

**Tip:** Always break absolute value inequalities into two separate cases for a complete solution.

Learn how to solve the absolute value inequality 4|x + 4| + 6 > 18 with a clear, step-by-step explanation. This example covers essential algebraic concepts like absolute value inequalities and interval notation. Suitable for students in Grades 8-10.

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