If $n$ is even ($n = 2k$), then $\text{Med}(Y)$ must lie in the closed interval $[x_k, x_{k+1}]$, where $x_k$ and $x_{k+1}$ are the elements adjacent to the original median.

If $n$ is odd ($n = 2k+1$), then $\text{Med}(Y)$ must lie in the closed interval $\left[ \frac{x_k + x_{k+1}}{2}, \frac{x_{k+1} + x_{k+2}}{2} \right]$.

The new median $\text{Med}(Y)$ is always strictly greater than the original median of $X$, regardless of the value of $y$.

If $y$ is an extreme outlier such that $y \to \infty$, then the new median $\text{Med}(Y)$ also approaches $\infty$.