Solve the following systems of inequalities graphically 2x + y ≥ 4, x + y ≤ 3, 2x – 3y ≤ 6.

x + y ≤ 3 ………… (2)

2x – 3y ≤ 6 ………… (3)

Graph of inequality (i): Let us draw the graph of the line 2x + y = 4.

y = 0 ⇒ 2x + 0 = 4 ⇒ x = 2.

x = 0 ⇒ 2(0) + y = 4 ⇒ y = 4.

(2, 0) and (0, 4) are on the line 2x + y ≥ 4.

Putting x = y = 0 in (i) we get 0 ≥ 4 which is false.

Hence, half-plane region not containing the origin is the solution region.

Graph of inequality (ii): Let us draw the graph of the line.

x + y = 3

y = 0 ⇒ x + 0 = 3 ⇒ x = 3.

x = 0 ⇒ 0 + y = 3 ⇒ y = 3.

Putting x = y = 0 in (ii) we get 0 ≤ 3 which is true.

Hence, half-plane region containing the origin is the solution region.

Graph of inequality (iii): Let us draw the graph of the line.

2x – 3y = 6

y = 0 ⇒ 2x + 3(0) = 6 ⇒ x = 3.

x = 0 ⇒ 2(0) – 3y = 6 ⇒ y = -2.

(3, 0) and (0, -2) are the points on the line 2x – 3y = 6.

Putting x = y = 0 in (ii) we get 0 ≤ 6 which is true.

Hence, half-plane region containing the origin is the solution region.

The common region of the above three regions is the solution set.