Solve : `|2x - 1| gt 2`

Explanation

To solve the inequality \( |2x - 1| > 2 \), we will break it down into two cases based on the definition of absolute value.

### Step 1: Set up the two cases for the absolute value inequality
The expression \( |A| > B \) can be rewritten as two separate inequalities:
1. \( A > B \)
2. \( A < -B \)

In our case, \( A = 2x - 1 \) and \( B = 2 \). Thus, we have:
1. \( 2x - 1 > 2 \)
2. \( 2x - 1 < -2 \)

### Step 2: Solve the first inequality \( 2x - 1 > 2 \)
To solve \( 2x - 1 > 2 \):
- Add 1 to both sides:
\[ 2x > 3 \]
- Divide both sides by 2:
\[ x > \frac{3}{2} \]

### Step 3: Solve the second inequality \( 2x - 1 < -2 \)
To solve \( 2x - 1 < -2 \):
- Add 1 to both sides:
\[ 2x < -1 \]
- Divide both sides by 2:
\[ x < -\frac{1}{2} \]

### Step 4: Combine the results
From the two inequalities, we have:
1. \( x > \frac{3}{2} \)
2. \( x < -\frac{1}{2} \)

### Step 5: Write the solution in interval notation
The solution can be expressed in interval notation as:
\[ x \in \left(-\infty, -\frac{1}{2}\right) \cup \left(\frac{3}{2}, \infty\right) \]

### Final Answer
The solution to the inequality \( |2x - 1| > 2 \) is:
\[ x \in \left(-\infty, -\frac{1}{2}\right) \cup \left(\frac{3}{2}, \infty\right) \]