Q.Show the graphical solution of the inequation $2x+3y\le 6$ in the coordinate plane.

Draw the boundary line through its intercepts (solid, since $\le$), then test the origin.

Step 1 — Boundary line. The related equation is $2x+3y=6$. Its intercepts: put $y=0\Rightarrow 2x=6\Rightarrow x=3$, giving $(3,0)$; put $x=0\Rightarrow 3y=6\Rightarrow y=2$, giving $(0,2)$. Join these two points.

Step 2 — Solid or dashed. The sign is $\le$ (weak), so the boundary line is solid and is part of the solution.

Step 3 — Test point. The line does not pass through the origin ($c=6\ne0$), so test $(0,0)$:

Step 4 — Shade. Since the origin satisfies the inequation, shade the half-plane containing the origin (the region below-left of the line). Together with the solid line, this shaded region is the graphical solution. …

Drawing the line dashed (that is only for strict $<,>$), or shading the wrong side by not actually substituting the test poi …