How many ordered pairs $(x,y)$ of positive integers satisfy the inequality $4x+5y < 20$?
We look for the number of points in the triangle marked by the -axis, -axis, and the line . We consider the rectangle with the vertices (0,0), (5,0), (0,4), and (5,4) (cutting this rectangle diagonally results in our triangle). The points contained in the 4x5 rectangle are a 3x4 grid of points, or 12 points in total. Since none of the points lie on the diagonal of the rectangle and the triangle is half of the rectangle, of the 12 points are contained in the triangle.