The maximum an complex expression If Z is a complex number such that :
$|Z+4|\leq3$ , find the maximum value of $|Z+1|$
My turn:
$|(x+4) + yi|\leq 3$ 
represents the surface of acircle whose center is (-4 ,0) and its radius is 3 , then
$-7\leq x \leq -1$ , then
$-6 \leq (x+1)\leq 0$
Then
$0\leq(x+1)^2 \leq 36 $
And
$-3\leq y \leq 3$
Then
$0\leq y^2 \leq 9$
Then
$0\leq (x+1)^2 + y^2 \leq 45$
Then
$|Z +1|\leq 3\sqrt{5}$
Then the required maximum value is $3\sqrt{5}$
Is the solution correct ?


As seen from the graph, the maximum $|z+1|$ is at $z=-7$, i.e.
$$|z+1| \le |-7+1| = 6$$
which can also be shown analytically as follows,
$$|z+1|=|z+4+(-3)| \le |z+4| + |-3| \le 3 +3 = 6$$
