Answer
The solution set is the region shaded GRAY
.
Work Step by Step
We will graph the solution region for each inequality.
1. (red) $(x-1)^{2}+(y+1)^{2}=5^{2}$
is a circle centered at $(1,-1)$, radius = $5$.
Border: dashed line ( sign is $< )$,
Testing $(1,-1):\quad 0^{2} + 0^{2} < 25\quad?$
Yes, shade the region containing $(1,-1).$
1. (blue) $(x-1)^{2}+(y+1)^{2}=4^{2}$
is a circle centered at $(1,-1)$, radius = $4$.
Border: solid line ( sign is $\geq )$,
Testing $(1,-1):\quad 0^{2} + 0^{2} \geq 9\quad?$
No, shade the region not containing $(1,-1).$
The solution set is the region with BOTH shadings.
The solution set is the region shaded GRAY