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.