Q.Graph the system of linear inequalities $x + y \le 6$, $x \ge 0$, $y \ge 0$, and shade the feasible region.

Graph each inequality's boundary and half-plane on the same axes, then shade only the region common to all three — this is the feasible region.

The boundary lines. $x+y=6$ meets the x-axis at $(6,0)$ and the y-axis at $(0,6)$; $x=0$ is the y-axis itself, and $y=0$ is the x-axis itself.

Testing each half-plane. For $x+y\le6$: testing the origin, $0+0=0\le6$ is true, so the origin's side of this line is shaded. The restrictions $x\ge0$ and $y\ge0$ simply restrict attention to the first quadrant (right of the y-axis and above the x-axis). …

Students sometimes shade the union of the three half-planes (a much larger, unbounded area) instead of their intersection, or forget that $x\ge0,y\ge0$ already excludes …