Let's solve the inequality:  
\[
4 \cdot |x + 4| + 6 > 18
\]

### Step 1: Isolate the absolute value expression
First, subtract 6 from both sides:
\[
4 \cdot |x + 4| > 12
\]

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

### Step 3: Split into two cases (definition of absolute value)
The expression \( |x + 4| > 3 \) translates into two inequalities:
1. \( x + 4 > 3 \)
2. \( x + 4 < -3 \)

### Step 4: Solve both inequalities

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

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

### Step 5: Combine the solutions
The solution to the inequality \( |x + 4| > 3 \) is:
\[
x > -1 \quad \text{or} \quad x < -7
\]

### Step 6: Write the final solution in interval notation
\[
x \in (-\infty, -7) \cup (-1, \infty)
\]

This is the solution to the original inequality.

---

Do you want any clarifications or further details?  
Here are five related questions:
1. What if the inequality was \( |x + 4| < 3 \)?
2. How does the solution change if we replace \( > \) with \( \geq \)?
3. Can you solve a similar problem with two absolute values, like \( |x - 2| + |x + 3| > 7 \)?
4. What happens if the absolute value is multiplied by a negative number?
5. How do we handle absolute value inequalities when dealing with quadratic terms?

**Tip:** Always be careful with the direction of inequalities when working with absolute values—splitting into two cases is essential!

Learn how to solve the absolute value inequality 4∣x + 4∣ + 6 > 18 with a clear step-by-step solution. This problem covers key algebraic concepts like absolute value inequalities, and isolating the absolute value expression. Suitable for students in Grades 9-10.

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