Answer
a. The graph of $\color{blue}{\vert z \vert =2}$ on the $xy$-plane is a circle centered at the origin and a having radius of 2.
b. The graph of $\color{brown}{\vert z \vert < 2}$ on the $xy$-plane is the interior of the circle centered at the origin and having a radius of 2.
c. The graph of $\color{green}{\vert z \vert > 2}$ on the $xy$-plane is the region exterior to the circle centered at the origin and having a radius of 2.
Work Step by Step
a.
$\begin{align*}
\vert z \vert &= 2 \\
\vert x+iy \vert &= 2 \\
\sqrt{x^2+y^2} &= 2 \\
\color{blue}{x^2+y^2}\ &\color{blue}{= 4}
\end{align*}$
The graph of $\color{blue}{\vert z \vert =2}$ on the $xy$-plane is a circle centered at the origin and a having radius of 2.
b.
$\begin{align*}
\vert z \vert &< 2 \\
\vert x+iy \vert &< 2 \\
\sqrt{x^2+y^2} &< 2 \\
\color{brown}{x^2+y^2}\ &\color{brown}{< 4}
\end{align*}$
The graph of $\color{brown}{\vert z \vert < 2}$ on the $xy$-plane is the interior of the circle centered at the origin and having a radius of 2.
c.
$\begin{align*}
\vert z \vert &> 2 \\
\vert x+iy \vert &> 2 \\
\sqrt{x^2+y^2} &> 2 \\
\color{green}{x^2+y^2}\ &\color{green}{> 4}
\end{align*}$
The graph of $\color{green}{\vert z \vert > 2}$ on the $xy$-plane is the region exterior to the circle centered at the origin and having a radius of 2.
