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The inequality provided is:

\[
|3x - 1| \geq 2
\]

This is an absolute value inequality. To solve it, we break it into two cases:

### Case 1: Positive inequality
\[
3x - 1 \geq 2
\]
Solving this:
\[
3x \geq 3 \quad \Rightarrow \quad x \geq 1
\]

### Case 2: Negative inequality
\[
-(3x - 1) \geq 2 \quad \Rightarrow \quad 3x - 1 \leq -2
\]
Solving this:
\[
3x \leq -1 \quad \Rightarrow \quad x \leq -\frac{1}{3}
\]

### Final Solution:
The solution to the inequality is the union of both cases:
\[
x \leq -\frac{1}{3} \quad \text{or} \quad x \geq 1
\]
In interval notation, this is:
\[
x \in \left( -\infty, -\frac{1}{3} \right] \cup \left[ 1, \infty \right)
\]

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

### Related Questions:
1. What is the general process for solving absolute value inequalities?
2. How do we interpret the solution of inequalities on a number line?
3. What happens when the inequality involves a less-than sign, e.g., \( |3x - 1| \leq 2 \)?
4. Can absolute value inequalities have no solutions? In what cases?
5. How can we check if our solution is correct by substituting values?

### Tip:
When solving absolute value inequalities, always split into two cases: one positive and one negative. This ensures you cover all possible scenarios for the solution.

Learn how to solve the absolute value inequality |3x - 1| ≥ 2 step-by-step. Understand the process of breaking down absolute value expressions into positive and negative cases and solve for the variable. This problem is suitable for students in Grades 9-11.

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