See figure and explanations.
Work Step by Step
Step 1. Find the solutions of $x^2+y^2\leq4$ as indicated by the red shaded area. Step 2. Find the solutions of $x+y\gt1$ as indicated by the blue shaded area. Step 3. The solutions for the system of inequalities can be found as the overlap of the above solutions, shown with the purple region (labeled solution region) in the figure.