|x-2| is greater than or equal to 1. solve the absolute value inequality.

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.

Learn how to solve the absolute value inequality |x - 2| ≥ 1 step-by-step. Understand the cases involved, the methods to remove absolute values, and the final solution expressed in interval notation. Suitable for Grades 9-12.

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