Q.Graph the region represented by the linear inequality $4x + y \ge 4$.

Apply the boundary-line-and-test-point method: draw $4x+y=4$, then test the origin in $4x+y\ge4$.

Intercepts: $(1,0)$ and $(0,4)$. Testing the origin: $4(0)+0=0$, and $0\ge4$ is false, so the shaded region is the side not containing the origin, boundary solid since the inequality is non-strict.

Boundary line. $4x+y=4$ meets the x-axis at $(1,0)$ (putting $y=0$) and the y-axis at $(0,4)$ (putting $x=0$).

Test point. The line does not pass through the origin, so test $(0,0)$ in $4x+y\ge4$: $4(0)+0=0$, and $0\ge4$ is false. So the required region is the half-plane not containing the origin.

Solid or dashed. The inequality is non-strict ($\ge$), so the boundary is drawn solid, included in the region.

Students sometimes shade the origin's side purely out of habit (since the origin worked as the correct side in earlier, $\le$-type examples), without actually re-testing it for this $\ge$ inequality.