Question

If $\left|z+4\right|\le 3$, then the greatest and least value of $\left|z+1\right|$ are

• A. $6,-6$
• B. $6,0$
• C. $7,2$
• D. $0,-1$

Answer 1

The correct option is B

$6,0$

Explanation for the correct option.

Find the range of the value of $\left|z+1\right|$.

It is given that $\left|z+4\right|\le 3$, then the compound inequality can be formed as:

$-3\le z+4\le 3$

Now, subtract $3$ from all sides.

$$\begin{gathered} -3-3\le z+4-3\le 3-3 \\ \Rightarrow -6\le z+1\le 0 \end{gathered}$$

Thus the modulus of $\left|z+1\right|$will lie between

$0\le \left|z+1\right|\le 6$.

Thus the least value of $\left|z+1\right|$ is $0$ and its greatest value is $6$.

Hence, the correct option is B.