To solve the absolute value inequality $|x - 2| \geq 1$, we'll break it down into two separate cases:

Case 1: $x - 2 \geq 1$

For this case, we directly remove the absolute value: $x - 2 \geq 1$ Add 2 to both sides: $x \geq 3$

Case 2: $x - 2 \leq -1$

For this case, we also remove the absolute value, but we flip the inequality and change the sign: $x - 2 \leq -1$ Add 2 to both sides: $x \leq 1$

Final Answer:

The solution to the inequality $|x - 2| \geq 1$ is: $x \leq 1 \quad \text{or} \quad x \geq 3$

This can be written in interval notation as: $(-\infty, 1] \cup [3, \infty)$

Would you like more details on this solution, or do you have any questions?

Here are 5 related questions you might find interesting:

1. How would the solution change if the inequality were $|x - 2| > 1$?
2. What happens if the inequality is $|x - 2| \leq 1$ instead?
3. Can you graph the solution to $|x - 2| \geq 1$ on a number line?
4. How do you solve $|2x - 4| \geq 3$ using a similar method?
5. What is the difference between solving an inequality with $\geq$ and $>$?

Tip: Always remember to split absolute value inequalities into two cases, reflecting the positive and negative scenarios.