# Solve the given inequality graphically in two- dimension. $2x +y \geq 6$

Toolbox:
• Same Quantity can be added (a subtracted ) to (from ) both sides of the inequality with out changing the sign of the in equality.
• Same positive quantities can be multiplied or divided to both side of the in equality with out changing the sign of the inequality.
• If same negative quantity is multiplied or divided to both sides of the inequality is reversed i.e $'>'$ sign changes to $'<'$ and $'<'$ changes $'>'$ .
• To represent the solution of linear inequality of one or two variable in a plane if the inequality involves $'\geq'$ or $' \leq$ we draw the graph of the line as a thick line to indicate the line is included in the solution set.
• If the inequality involves $'>'$ as $'<'$ we draw the graph of the line using is not included in the solution set.
• We consider the corresponding equation $ax+by =c$ which represents a straight line This line divides the plane into two half planes I and II
• We take any point in I half plane and check if it satisfies the given inequality will be one half plane (called solution region ) Containing the point satisfying the inequality
Step 1:
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
Step 2:
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
Step 3:
The solution region of the given inequality is the shaded half including the line.