We are given the inequality:

$2|x - 2| + 6 \geq 26$

Step 1: Isolate the absolute value term.

Subtract 6 from both sides:

$2|x - 2| \geq 20$

Step 2: Divide by 2.

$|x - 2| \geq 10$

Step 3: Solve the absolute value inequality.

The absolute value inequality $|x - 2| \geq 10$ means that:

$x - 2 \leq -10 \quad \text{or} \quad x - 2 \geq 10$

Step 4: Solve each part.

1. $x - 2 \leq -10$: $x \leq -8$

2. $x - 2 \geq 10$: $x \geq 12$

Final Solution:

The solution to the inequality is:

$x \leq -8 \quad \text{or} \quad x \geq 12$

In interval notation, this is:

$(-\infty, -8] \cup [12, \infty)$

Would you like further details or explanations?

Here are 5 related questions to expand your understanding:

1. How do you solve absolute value equations?
2. What is the difference between solving absolute value inequalities with "<" and ">”?
3. How can you graph the solution to an absolute value inequality?
4. What are the steps to solve compound inequalities?
5. How do you check if a value satisfies an absolute value inequality?

Tip: When solving absolute value inequalities, always remember that $|a| \geq b$ means $a \leq -b$ or $a \geq b$.