The given inequality is:

$4|x + 3| - 2 \geq 18$

Step-by-Step Solution:

1. Isolate the absolute value: $4|x + 3| - 2 \geq 18 \Rightarrow 4|x + 3| \geq 20$

2. Divide by 4 on both sides: $|x + 3| \geq 5$

3. Interpret the absolute value inequality: For $|x + 3| \geq 5$, this means that the expression inside the absolute value is either: $x + 3 \geq 5 \quad \text{or} \quad x + 3 \leq -5$

4. Solve each case:

   • Case 1: $x + 3 \geq 5 \Rightarrow x \geq 2$
   • Case 2: $x + 3 \leq -5 \Rightarrow x \leq -8$

Final Answer:

The solution is:

$x \geq 2 \quad \text{or} \quad x \leq -8$

In interval notation, the solution set is:

$(-\infty, -8] \cup [2, \infty)$

Let me know if you'd like further details or have any questions!

Related Questions:

1. How do you solve absolute value equations instead of inequalities?
2. What is the graphical representation of this solution on the number line?
3. How would the inequality change if the direction was reversed?
4. Can absolute value inequalities have "no solution" cases?
5. How does adding or subtracting values inside the absolute value impact the solution?

Tip: When solving absolute value inequalities, remember to consider the two cases separately to capture all possible solutions!