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.

Explore the solution to the absolute value inequality 4|x + 4| + 6 > 18. This step-by-step solution covers algebraic concepts, such as breaking inequalities into two cases and interval notation. Suitable for students in Grades 9-12, this problem builds a foundation in solving inequalities involving absolute values.

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