Question

If $\left| {z + 4} \right| \leqslant 3$ , then what is the greatest and the least value of $\left| {z + 1} \right|$ ?
A) -1, 6
B) 6, 0
C) 6, 3
D) None of the above

Answer 1

Hint: The modulus function is an important function that provides us with the absolute value of a number or a variable. If $\left| x \right| = a$ , then $x = \pm a$ . In inequality, this function has important properties. For the above question use $\left| {{z_1} - {z_2}} \right| \leqslant \left| {{z_1}} \right| + \left| {{z_2}} \right|$ to obtain the desired result.

Complete step by step Solution:
It is given that: $\left| {z + 4} \right| \leqslant 3$ … (1)
Now, we can also write $\left| {z + 1} \right| = \left| {(z + 4) - 3} \right|$ .
As by the property of modulus function, we know that $\left| {{z_1} - {z_2}} \right| \leqslant \left| {{z_1}} \right| + \left| {{z_2}} \right|$ .
Hence,
$\left| {(z + 4) - 3} \right| \leqslant \left| {z + 4} \right| + \left| 3 \right|$
From equation (1), we know that $\left| {z + 4} \right| \leqslant 3$ .
Hence, using that in the above equation, we get:
$\left| {z + 1} \right| \leqslant 3 + \left| 3 \right|$
Therefore,
$\left| {z + 1} \right| \leqslant 6$
This implies that the greatest value of $\left| {z + 1} \right|$ is 6.
Now, as the value of the modulus function cannot be negative,
This implies that the least value of $\left| {z + 1} \right|$ is 0.
Hence, the greatest and the least values are 6 and 0 respectively.

Therefore, the correct option is B.

Note:The modulus function gives us the absolute value of the number of variables within it. As it gives us the absolute value, the result of the modulus function can never give us a negative result. Hence, the minimum value of the modulus function will always be 0.