If $|z+4|\le 3$, then the maximum value of $|z+1|$ is:

We want to find the maximum value of $|z+1|$.Using the triangle inequality $|a+b|\le |a|+|b|$:
$$|z+1|=|z+4-3|$$
$$|z+1|\le |z+4|+|-3|$$
Given $|z+4|\le 3$:
$$|z+1|\le 3+3=6$$
The maximum value is $6$.Geometrically, $z$ lies in a circle with center $(-4,0)$ and radius $3$. The point $z+1$ represents the distance from $z$ to $(-1,0)$. The farthest point in the disk from $(-1,0)$ is along the diameter, which is $3+3=6$.