For \(x+4y\geq 12\), after testing \((0,0)\), which side is shaded?

Correct answer: opposite side of origin

Substituting \((0,0)\) gives \(0+4(0)\geq 12\), i.e. \(0\geq 12\), which is false. Hence, the half-plane containing the origin is not shaded; the half-plane opposite to the origin is shaded. The boundary line \(x+4y=12\) is included because the inequality has \(\geq\). Exam tip: if a test point makes the inequality false, shade the side opposite to that point.