See figure and explanations. The solution is the purple region in the positive x and y regions ($x\geq0$ and $y\geq0$).
Work Step by Step
Step 1. Conditions $x\geq0$ and $y\geq0$ define the solutions to be in the first quadrant. Step 2. Find the solutions of $2x+y\lt4$ as indicated by the red shaded area (turned into purple after overlapping with blue region). Step 3. Find the solutions of $2x-3y\leq6$ as indicated by the blue shaded area. Step 4. The solutions for the system of inequalities can be found as the overlap of the above solutions, shown with the purple region in the first quadrant (labeled solution region) in the figure.