The solutions to the inequalities 0≤x≤2,0≤y≤1, and y≤−12x+32 form a polygonal region in the plane. how many sides does this polygon have?

In order to know the shape of the polygon you plot first the last inequality because it form a line in the graph. In plotting this equation, disregard the inequality sign first. Then, set arbitrary points of x, determine y and plot them. The line is shown in the picture. Next, we test the inequality. Suppose we choose the point of origin, then,

y≤−12x+32
0 ≤ -12(0)+32
0 ≤ 32

This is true. Thus, the shaded region must include (0,0) which is located to the left of line (blue-shaded region). Next, use the two remaining boundaries. The inequality 0≤x≤2 is denoted as the black horizontal line from x=0 to x=2. Moreover, the inequality 0≤y≤1 is denotes as the black vertical line. The resulting figure is a rectangle. Therefore, the polygon has 4 sides.