The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ is contained entirely in quadrants:
$\textbf{(A)}\ \text{I and II}\qquad \textbf{(B)}\ \text{II and III}\qquad \textbf{(C)}\ \text{I and III}\qquad \textbf{(D)}\ \text{III and IV}\qquad \textbf{(E)}\ \text{I and IV}$
Graph both equations.  The solution to the system of inequalities is the magenta area because it is in both of the solutions of each individual inequality.  From the graph, the magenta area is in quadrants $I$ and $II$, so the answer is $\boxed{\textbf{(A)}}$.