Graphical Representation of Linear Inequalities

A straight line $ax+by=c$ splits the Cartesian plane into two half-planes; a vertical line $x=c$ gives left/right half-planes and a horizontal line $y=k$ gives upper/lower half-planes. Any point NOT on the line lies in exactly one of the two half-planes, and satisfies exactly one of $ax+by<c$ or $ax+by>c$.

Test-point method. To identify which half-plane an inequality like $ax+by<c$ represents: pick a convenient point $P$ not on the line (the origin, when $c\ne0$), and substitute its coordinates into the inequality. If it holds, the required half-plane is the one containing $P$; if not, it is the one not containing $P$.

Worked pattern. For $x\ge2$: the line $x=2$ splits the plane; substituting $(0,0)$ gives $0\ge2$, FALSE, so the shaded region is the side that does NOT contain the origin (points with $x\ge2$), including the boundary line itself since the inequality is non-strict. Similarly for $x+2y>3$: substituting the origin gives $0>3$, FALSE, so the solution is the half-plane not containing the origin. …