Question

If $\left|z+4\right|\le 3$, then find the greatest and least values of $\left|z+1\right|$.

Answer 1

$$\begin{gathered} \left|z+1\right|=\left|z+4-3\right| \\ \le \left|z+4\right|+\left|-3\right| \\ \le 3+3 \\ =6 \\ \mathrm{Also}, \\ \left|z+1\right|\ge 0 \\ \mathrm{Thus},0\le \left|z+1\right|\le 6. \end{gathered}$$

Hence, the greatest and least values of $\left|z+1\right|$ is 6 and 0.