Let's solve the inequality:

$4|x+4| + 6 > 18$

Step 1: Isolate the absolute value term

First, subtract 6 from both sides:

$4|x+4| > 18 - 6$

This simplifies to:

$4|x+4| > 12$

Step 2: Divide by 4

Now, divide both sides by 4:

$|x+4| > 3$

Step 3: Break into two cases

Since we are dealing with an absolute value inequality, we have two cases:

1. Case 1: $x + 4 > 3$

   Subtract 4 from both sides:

   $x > -1$

2. Case 2: $x + 4 < -3$

   Subtract 4 from both sides:

   $x < -7$

Step 4: Combine the results

The solution is the union of both cases:

$x > -1 \quad \text{or} \quad x < -7$

In interval notation, the solution is:

$(-\infty, -7) \cup (-1, \infty)$

Do you want further details or have any questions about this solution?

Here are five related questions for further practice:

1. Solve $3|x-2| - 5 < 10$.
2. Solve $|x + 7| + 3 \leq 10$.
3. Find the solution to $5|x - 1| + 4 > 9$.
4. Solve $2|2x + 1| - 3 \geq 1$.
5. Solve $|x-3| - 2 < 4$.

Tip: Always break absolute value inequalities into two cases to solve them accurately.