Solve : 3 x - 6 ≥ 0 graphically in two dimensional plane.

We have 3 x - 6 ≥ 0 ⇒ 3 x - 2 ≥ 0 ⇒ x - 2 ≥ 0 Consider the equation x - 2 = 0 ⇒ x = 2 But, x = 2 represents a line parallel to y -axis at a distance of 2 units from it. We draw the line x = 2 , making it as a solid line. If we put x = 0 in the given inequation, we get - 2 ≥ 0 , which is false i.e. x = 0 does not satisfy the given inequation. So, solution region consists of the shaded region on the right-hand side of the line x = 2 including the line x = 2 , as shown below:

Solve : 2 x + y ≤ 6 graphically.

We have, 2 x + y ≤ 6 Considering equality 2 x + y = 6 ⇒ x 3 + y 6 = 1 Hence, it cuts the x - axis at A 3 , 0 and y - axis at B 0 , 6 . Origin is satisfying given inequality, hence required solution region is

Solve: 5 x + 4 y ≤ 40 , x ≥ 2 and y ≤ 3 graphically.

We have, 5 x + 4 y ≤ 40 x ≥ 2 y ≤ 3 Let L : 5 x + 4 y = 40 L cuts x - axis at A 8 , 0 and y - axis at B 0 , 10 . Origin is satisfying the given inequality, hence required solution region is