Solve the system of inequalities graphically x + 2y ≤ 10, x + y ≥ 1, x – y ≤ 0, x ≥ 0, y ≥ 0.

x + y ≥ 1 ………………. (2)

x - y ≤ 0 ……………… (3)

x ≥ 0 ……………… (4)

y ≥ 0 ……………… (5)

Graph of inequality (i). Let us draw the graph of the line x + 2y ≤ 10.

y = 0 ⇒ x + 2(0) = 10 ⇒ x = 10

x = 0 ⇒ 0 + 2y ⇒ y = 5

(10, 0) and (0, 5) are the points on the line x + 2y = 10.

Putting x = y = 0 in (i) we have 0 ≤ 10 which is true. Hence half-plane region containing the origin is the solution region.

Graph of inequality (ii). Let us draw the graph of the line x + y = 1.

At y = 0 ⇒ x + (0) = 1 ⇒ x = 1

x = 0 ⇒ 0 + y = 1 ⇒ y = 1

∴ (1, 0) and (0, 1) are the points on the line x + y = 1.

Putting x = y = 0 in (ii) we have 0 ≥ 1 which is false.

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

Graph of inequality (iii). Let us draw the graph of the line x - y = 0.

On the line

At x = 0 ⇒ 0 – y = 0 ⇒ y = 0

at x = 1 ⇒ 1 – y = 0 ⇒ y = 1

∴ (0, 0) and (1, 1) are on the line x – y = 0.

To determine the region represented by the given inequality (iii) consider the point not lying on the line x – y = 0 say (2, 0) and it his in the half-plane of (iii) if 2 ≤ which is not true. Therefore the portion not containing (2, 0) represents the solution set of the given inequality.

Graph of inequality (iv) and (v).

Clearly, x ≥ 0, represents the region lying on the right side of y-axis and y ≥ 0 represents the region lying above the x-axis.

The common region of the above five regions is the solution sets.