Solve `|x-1|+|x-2| ge 4`

Explanation

To solve the inequality \( |x-1| + |x-2| \geq 4 \), we will analyze the expression by breaking it down into different cases based on the values of \( x \).

### Step 1: Identify critical points
The critical points for the absolute values are \( x = 1 \) and \( x = 2 \). We will consider three cases based on these points:
1. \( x < 1 \)
2. \( 1 \leq x < 2 \)
3. \( x \geq 2 \)

...