See figure and explanations.
Work Step by Step
Step 1. Find the solutions of $x+y\gt4$ as indicated by the upper right shaded area (originally red but turned to purple after overlapping with the blue region). Step 2. Find the solutions of $x+y\gt-1$ 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.