*Question 1*

*Solve the system of inequalities graphically: *

*= x **≥** 3 and y **≥** 2*

**Sol:**

= x ≥ 3 ……………………(i)

= y ≥ 2……………………..(ii)

In order to draw the graph of the following inequalities, let us consider x = 3 and y = 2

In case of equation (i) the feasible region is the right hand side of the line including the points on the line x = 3 , whereas in case of equation (ii) the feasible region is above the line including the points on the line y = 2.

Therefore, the solution of the given set of linear inequalities is the common shaded area including the points on the line as shown in the graph below.

*Question 2*

*Solve graphically the system of inequalities: *

*3x + 2y **≤** 12, x **≥** 1 and y **≥** 2*

**Sol:**

3x + 2y ≤ 12 …………… (i)

x ≥ 1 ……………………… (ii)

y ≥ 2 ……………………….(iii)

In order to draw the graph of the following inequalities, let us consider 3x + 2y = 12, x = 1 and y = 2

In case of equation (i) the feasible region is below the line including the points on the line 3x + 2y = 12. In case equation (ii) the feasible region is on the right hand side including the points on the line x = 1 and in case of equation (iii) the feasible region is above the line including the points on the line y = 2.

Therefore, the solution of the given set of linear inequalities is the common shaded area including the points on the line as shown in the graph below.

*Question 3*

*Solve graphically the system of inequalities: *

*2x + y **≥** 6 and 3x + 4y **≤** 12.*

**Sol:**

2x + y ≥ 6 …………………………. (i)

3x + 4y ≤ 12……………………… (ii)

In order to draw the graph of the following inequalities, let us consider 2x + y = 6 and 3x + 4y = 12.

In case of equation (i) the feasible region is above the line including the points on the line 2x + y = 6 , whereas in case of equation (ii) the feasible region is below the line including the points on the line 3x + 4y = 12

Therefore, the solution of the given set of linear inequalities is the common shaded area including the points on the line as shown in the graph below.

*Question 4*

*Solve graphically the system of inequalities: *

*x + y **≥** 4 and 2x – y **˃** 0*

**Sol:**

x + y ≥ 4 …………………………. (i)

2x – y ˃ 0……………………… (ii)

In order to draw the graph of the following inequalities, let us consider x + y = 4 and 2x – y = 0.

In case of equation (i) the feasible region is above the line including the points on the line x + y = 4, whereas in case of equation (ii) the feasible region is the right hand side of the line including the points on the line 2x – y = 0.

*Question 5*

*Solve graphically the system of inequalities: *

*2x – y **˃** 1 and x – 2y **˂** -1*

**Sol:**

2x – y ˃ 1 …………………………. (i)

x – 2y ˂ -1……………………… (ii)

In order to draw the graph of the following inequalities, let us consider 2x – y = 1 and x – 2y = -1.

In case of equation (i) the feasible region is below the line including the points on the line 2x – y = 1, whereas in case of equation (ii) the feasible region is the above the line including the points on the line x – 2y = -1.

*Question 6*

*Solve graphically the system of inequalities: *

*x + y **≤** 6 and x + y **≥** 4*

**Sol:**

x + y ≤ 6 …………………………. (i)

x + y ≥ 4……………………… (ii)

In order to draw the graph of the following inequalities, let us consider x + y = 6 and x + y = 4.

In case of equation (i) the feasible region is below the line including the points on the line x + y = 6, whereas in case of equation (ii) the feasible region is the above the line including the points on the line x + y = 4.

*Question 7*

*Solve graphically the system of inequalities: *

*2x + y **≥** 8 and x + 2y **≥** 10*

**Sol:**

2x + y ≥ 8 …………………………. (i)

x + 2y ≥ 10……………………… (ii)

In order to draw the graph of the following inequalities, let us consider 2x + y = 8 and x + 2y = 10.

In case of equation (i) the feasible region is above the line including the points on the line 2x + y = 8 , whereas in case of equation (ii) the feasible region is the above the line including the points on the line x + 2y = 10.

*Question 8*

*Solve graphically the system of inequalities: *

*x + y **≤** 9, y **˃** x and x **≥** 0*

**Sol:**

x + y ≤ 9 …………………………. (i)

y ˃ x ……….……………………… (ii)

x ≥ 0 ………………………………… (iii)

In order to draw the graph of the following inequalities, let us consider x + y = 9, y = x and x = 0.

In case of equation (i) the feasible region is below the line including the points on the line x + y = 9. In case of equation (ii) the feasible region is above the line including the points on the line y =x , whereas in case of equation (iii) the feasible region is the right hand side of the line including the points on the line x = 0

*Question 9*

*Solve graphically the system of inequalities: *

*5x + 4y **≤** 20, x **≥** 1 and y **≥** 2*

**Sol:**

5x + 4y ≤ 20 …………………………. (i)

x ≥ 1 ……….……………………… (ii)

y ≥ 2 ………………………………(iv)

In order to draw the graph of the following inequalities, let us consider 5x + 4y = 20, x = 1, x = 0 and y = 2

In case of equation (i) the feasible region is below the line including the points on the line 5x + 4y = 20. In case of equation (ii) the feasible region is on the right hand side of the line including the points on the line x = 1, where as in equation (iv) the feasible region is above line including the points on the line y = 2

* *

*Question 10*

*Solve graphically the system of inequalities: *

*3x + 4y **≤** 60, x + 3y **≤** 30 and x **≥** 0*

**Sol:**

3x + 4y ≤ 60 …………………………. (i)

x + 3y ≤ 30 ……………………… (ii)

x ≥ 0 …………………………………….(iii)

In order to draw the graph of the following inequalities, let us consider 3x + 4y = 60, x + 3y = 30 and x = 0

In case of equation (i) the feasible region is below the line including the points on the line 3x + 4y = 60. In case of equation (ii) the feasible region is the below the line including the points on the line x + 3y = 30, whereas in case of equation (iii) the feasible region is right and side of the line.

*Question 11*

*Solve graphically the system of inequalities: *

*2x + y **≥** 4, x + y **≤** 3 and 2x -3y **≤** 6*

**Sol:**

2x + y ≥ 4 …………………………. (i)

x + y ≤ 3 ……….……………………… (ii)

2x – 3y ≤ 6 ………………………………… (iii)

In order to draw the graph of the following inequalities, let us consider 2x +y = 4, x + y = 3 and 2x – 3y = 6.

In case of equation (i) the feasible region is above the line including the points on the line 2x + y = 4. In case of equation (ii) the feasible region is below the line including the points on the line x + y = 3, whereas in case of equation (iii) the feasible region is above line including the points on the line 2x- 3y = 6

*Question 12*

*Solve graphically the system of inequalities: *

*5x + 4y **≤** 3, 3x + 4y **≥** 12, x **≥** 0 and y **≥** 1*

**Sol:**

x – 2y ≤ 3 …………………………. (i)

3x + 4y ≥ 12 ……….……………………… (ii)

x ≥ 0 ………………………………… (iii)

y ≥ 1 ………………………………(iv)

In order to draw the graph of the following inequalities, let us consider x – 2y = 3, 3x + 4y = 12, x = 0 and y = 1

In case of equation (i) the feasible region is above the line including the points on the line x – 2y = 3. In case of equation (ii) the feasible region is above the line including the points on the line 3x + 4y = 12, whereas in case of equation (iii) the feasible region is in the right hand side of the axis including the points on the line x = 0 and (iv) the feasible region is above line including the points on the line y = 1

*Question 13*

*Solve graphically the system of inequalities: *

*4x + 3y **≤** 60, y **≥** 2x, x **≥** 3 and x, y **≥** 0*

**Sol:**

4x + 3y ≤ 60 …………………………. (i)

y ≥ 2x ……….……………………… (ii)

x ≥ 3 ………………………………… (iii)

x ≥ 0 and y ≥ 0 …………………(iv)

In order to draw the graph of the following inequalities, let us consider 4x + 3y = 60, y = 2x, x = 3 and x, y = 0

In case of equation (i) the feasible region is below the line including the points on the line 4x + 3y = 60. In case of equation (ii) the feasible region is above the line including the points on the line y = 2x, whereas in case of equation (iii) the feasible region is in the right hand side of the axis including the points on the line x = 3 and (iv) the feasible region is above the line and right hand side of the line including the points on the line y = 0 and x =0 respectively.

*Question 14*

*Solve graphically the system of inequalities: *

*3x + 2y **≤** 150, x + 4y **≤** 80, x **≤** 15 *

**Sol:**

3x + 2y ≤ 150 …………………………. (i)

x + 4y ≤ 80 ……….……………………… (ii)

x ≤ 15 ………………………………… (iii)

In order to draw the graph of the following inequalities, let us consider 3x + 2y = 150, x + 4y = 80, x = 15.

In case of equation (i) the feasible region is below the line including the points on the line 3x + 2y = 150. In case of equation (ii) the feasible region is below the line including the points on the line x + 4y = 80 , whereas in case of equation (iii) the feasible region is in the left hand side of the axis including the points on the line x = 15.

*Question 15*

*Solve graphically the system of inequalities: *

*x + 2y **≤** 10, x + y **≥** 1, x – y **≤** 0 *

**Sol:**

x + 2y ≤ 10 …………………………. (i)

x + y ≥ 1 ……….……………………… (ii)

x – y ≤ 0 ………………………………… (iii)

In order to draw the graph of the following inequalities, let us consider x + 2y = 10, x + 4y = 1, x – y = 0.

In case of equation (i) the feasible region is below the line including the points on the line x + 2y = 10. In case of equation (ii) the feasible region is above the line including the points on the line x + y = 1, whereas in case of equation (iii) the feasible region is above the line including the points on the line x – y =0.