If a complex number $z$ lies in the interior or on the boundary of a circle of radius 3 units and centre $(-4{,}0)$, find the greatest and least values of $|z+1|$.

Step 1: Understand the given condition

The problem states that the complex number $z$ lies in the interior or on the boundary of a circle with radius 3 units and center $(-4{,}0)$. In the complex plane, the distance between a complex number $z$ and a complex number $c$ is given by $|z-c|$. Therefore, the condition can be written as $|z - (-4)| \le 3$, which simplifies to $|z+4| \le 3$. This means $z$ is a point within or on the circle centered at $(-4{,}0)$ with radius 3.

Step 2: Interpret the expression to be maximized/minimized

We need to find the greatest and least values of $|z+1|$. This expression represents the distance between the complex number $z$ and the complex number $-1$ (which corresponds to the point $(-1{,}0)$ in the complex plane).

Step 3: Visualize the geometry

Let the center of the given circle be $C = (-4{,}0)$ and its radius be $r=3$. Let the point corresponding to the complex number $-1$ be $P = (-1{,}0)$. We are looking for the maximum and minimum distances from any point $z$ within or on the circle to the point $P$.

Step 4: Calculate the distance between the center and the point P

First, calculate the distance $d$ between the center of the circle $C(-4{,}0)$ and the point $P(-1{,}0)$. This distance is $|-4 - (-1)| = |-3| = 3$. Notice that this distance is equal to the radius of the circle.

Step 5: Determine the greatest value of $|z+1|$

The greatest distance from point $P$ to any point $z$ on or inside the circle occurs when $z$ is on the circle, on the line passing through $P$ and $C$, and on the opposite side of $C$ from $P$. This maximum distance is the sum of the distance $d$ from $P$ to $C$ and the radius $r$. So, $|z+1|_{\text{max}} = d + r = 3 + 3 = 6$.

Step 6: Determine the least value of $|z+1|$

The least distance from point $P$ to any point $z$ on or inside the circle occurs when $z$ is on the circle, on the line passing through $P$ and $C$, and on the same side of $C$ as $P$. This minimum distance is the absolute difference between the distance $d$ from $P$ to $C$ and the radius $r$. Since $d=r=3$, the minimum distance is $3 - 3 = 0$. This means the point $P(-1{,}0)$ lies on the boundary of the circle, so $z$ can be equal to $-1$, making $|z+1|=0$.

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