The inequalities are
2x + y ≥ 6, 3x + 4y ≤ 12
(i) The line l1 : 2x + y = 6 passes through (3, 0), (0, 6)
AB represents the line 2x + y = 6
Putting x = 0, y = 0 in 2x + y ≥ 6
0 ≥ 6, which is false.
∴ Origin does not lie in the region of 2x + y ≥ 6
Therefore, the region lying above the line AB and all the points on AB represents the inequality 2x + y ≥ 6
(ii) The line l2 : 3x + 4y = 12 passes through (4, 0) and (0, 3).
This line is represented by CD.
Consider the inequality 3x + 4y ≤ 12
Putting x = 0, y = 0 in 3x + 4y ≤ 12, we get 0 ≤ 12, which is true.
∴ 3x + 4y ≤ 12 represents the region below the line CD (towards origin) and all the points lying on it.
The common region is the solution of 2x + 3y ≥ 6 are 3x + 4y ≤ 12 represented by the shaded region in the graph.