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.