Mathematics

The key is to find which values of $k$ allow the domain to be non-empty. Since we need both square roots to be defined, we need $x + k \geq 0$ AND $5 - 2x \geq 0$ simultaneously. This means $x \geq -k$ and $x \leq 5/2$, so the domain is non-empty exactly when these intervals overlap—that is, when $-k \leq 5/2$. Rearranging this inequality gives you the constraint on $k$.

The key is to find which values of $k$ allow the domain to be non-empty. Since we need both square roots to be defined, we need $x + k \geq 0$ AND $5 - 2x \geq 0$ simultaneously. This means $x \geq -k$ and $x \leq 5/2$, so the domain is non-empty exactly when these intervals overlap—that is, when $-k \leq 5/2$. Rearranging this inequality gives you the constraint on $k$.