To solve the inequality $2|x + 4| + 8 \geq 18$, follow these steps:

Step 1: Isolate the absolute value expression

First, subtract 8 from both sides: $2|x + 4| \geq 10$

Next, divide both sides by 2: $|x + 4| \geq 5$

Step 2: Break the absolute value into two inequalities

The definition of absolute value gives two possible cases:

1. $x + 4 \geq 5$
2. $x + 4 \leq -5$

Case 1: $x + 4 \geq 5$

Subtract 4 from both sides: $x \geq 1$

Case 2: $x + 4 \leq -5$

Subtract 4 from both sides: $x \leq -9$

Final Solution:

The solution is the union of both inequalities: $x \geq 1 \quad \text{or} \quad x \leq -9$ This can be written in interval notation as: $x \in (-\infty, -9] \cup [1, \infty)$

Would you like further details or have any questions?
Here are some related questions for practice:

1. Solve $3|x - 2| - 7 \geq 5$.
2. Solve $|x + 1| + 4 < 9$.
3. What is the solution to $4|x + 3| \leq 20$?
4. Solve $|2x - 5| + 6 = 10$.
5. Find the solution for $|x + 6| > 7$.

Tip: Always isolate the absolute value expression before breaking it into cases.