Answer
Please see the graph.
Work Step by Step
Red inequality: $x^{2}+\left(y-2\right)^{2}\ge9$
Blue inequality: $\frac{x^{2}}{4}+\frac{y^{2}}{25}<1$
$x^{2}+\left(y-2\right)^{2}\ge9$
This inequality is for a circle with radius 3, and this circle is shifted up two units (compared to a circle centered at the origin).
If a point is outside the circle, then it is part of the solution set.
$\frac{x^{2}}{4}+\frac{y^{2}}{25}<1$
This graph is for an ellipse. We pick the point $(0,0)$ to determine whether to shade inside the ellipse or outside the ellipse.
$\frac{x^{2}}{4}+\frac{y^{2}}{25}<1$
$\frac{0^{2}}{4}+\frac{0^{2}}{25}<1$
$\frac{0}{4}+\frac{0}{25}<1$
$0 + 0 < 1$
$0 < 1$ (true, so we shade inside the ellipse)
The solution set of the inequalities is the overlap of the two graphs.
