Answer
Please see the graph.
Work Step by Step
Orange graph: $x^{2}+y^{2}\ge4$
Red graph: $x^{2}+y^{2}<16$
Green graph: $y\ge0$
The orange and green graphs have greater than or equal to signs, so the graphs will have solid lines. The red graph has a less than sign, so that graph will have a dotted line.
The orange and red graphs are the equations for circles with radii 2 and 4, respectively.
We use the point $(0,0)$ to determine what sides of the graphs to shade.
$x^{2}+y^{2}\ge4$
$0^{2}+0^{2}\ge4$
$0 + 0 \ge 4$
$0 \ge 4$ (false, so we shade the side of the graph without the point)
$x^{2}+y^{2}\lt 16$
$0^{2}+0^{2}\lt 16$
$0 + 0 \lt 16$
$0 \lt 16$ (true, so we shade the side of the graph with the point)
$y\geq0$
$0 \geq 0$ (true, so we shade the side of the graph with the point)
The overlap of the three graphs is the solution set for the set of inequalities.
