I want to plot $|2z+3|\gt 4$. Firstly I plotted $|2z|\gt 4$, by taking it to be:$$|z|\gt 2$$ $$|x+iy|\gt 2$$
And this I am comfortable with, it is just everything greater than(in x or y values) than the line passing through $(0,2),(2,0)$.
Then I noted that, for example we can have $|z-2|\leq 1$ gives $|(x-2)+iy|\leq 1$ which is a circle centered at $(2,0)$, so this $-2$ translates my circle $2$ in the positive direction.
Now then I think that $|2z+3|\gt 4$ translates my line $3$ to the left. But then I tried just getting two points to be equal to $4$ to see where my line is, and I got the line passing through $(0,.5),(.5,0)$. Which aren't consistent.
How do I plot this domain? How should I think about these things?
$$\newcommand{\t}[1]{\text{#1}} |z-(-3/2)|\ge 2\\ \t{matches with}\\ |z-z_0|\ge r\\\iff |(x+iy)-(x_0+iy_0)|\ge r\iff|(x-x_0)+i(y-y_0)|\ge r\\ \iff \sqrt{(x-x_0)^2+(y-y_0)^2}\ge r\iff (x-x_0)^2+(y-y_0)^2\ge r^2 $$ Which is actually the exterior of circle with center $z_0$ and radius $r$, since distance from point is greater than r (think.think.)