Understand the condition for an absolute value equation to have a solution.

For an equation of the form $|X| = Y$ to have a solution, the right-hand side $Y$ must be non-negative, i.e., $Y \ge 0$. This is because the absolute value of any real number is always greater than or equal to zero.

The given equation is $|x-a| = 2+K$.
Here, $X = x-a$ and $Y = 2+K$.
For this equation to have a solution, we must have:

Subtract 2 from both sides of the inequality:

The values of $K$ that satisfy $K \ge -2$ are all real numbers from -2 upwards, including -2.
This is represented by the interval $[-2, +\infty[$.

A. $K \in ]-\infty; 2]$
B. $K \in [-2; +\infty[$
C. $K \in ]-\infty; a]$
D. $K \in [-...$ (partially visible)

The correct option is B.

The final answer is $\boxed{\text{B. } K \in [-2; +\infty[}$.