The given inequality is $2x+y \geq 6$
Consider the equation $2x+y=6$
We observe that $(3,0) $ and $(0,6)$
Satisfy the equation
The graphical representation of the line $2x+y=6$ is
The Line divides the xy- plane into two half plane I and II select a point not on the line say (0,0)
We see that $2(0) +0 \geq 6$ is false .
Therefore half plane I containing point (0,0) is not the solution of the given inequality
The solution region of the given inequality is the shaded half including the line.