Question 1: Solve the following systems of linear inequations graphically:

i) $2x+3y \leq 6 , \hspace{0.3cm} 3x+2y \leq 6, \hspace{0.3cm}x \geq 0, y \geq 0$

ii) $2x+3y \leq 6 , \hspace{0.3cm} x+4y \leq 4, \hspace{0.3cm}x \geq 0, y \geq 0$

iii) $x-y \leq 1 , \hspace{0.3cm} x+2y \leq 8, \hspace{0.3cm} 2x+y \geq 2, \hspace{0.3cm}x \geq 0, y \geq 0$

iv) $x+y \geq 1 , \hspace{0.3cm} 7x+9y \leq 63, \hspace{0.3cm} x \leq 6, \hspace{0.3cm} y \leq 5, \hspace{0.3cm}x \geq 0, y \geq 0$

v) $2x+3y \leq 35 , \hspace{0.3cm} y \geq 3, \hspace{0.3cm} \hspace{0.3cm}x \geq 2, x \geq 0, y \geq 0$

i) $2x+3y \leq 6 , \hspace{0.3cm} 3x+2y \leq 6, \hspace{0.3cm}x \geq 0, y \geq 0$ ii) $2x+3y \leq 6 , \hspace{0.3cm} x+4y \leq 4, \hspace{0.3cm}x \geq 0, y \geq 0$ iii) $x-y \leq 1 , \hspace{0.3cm} x+2y \leq 8, \hspace{0.3cm} 2x+y \geq 2, \hspace{0.3cm}x \geq 0, y \geq 0$ iv) $x+y \geq 1 , \hspace{0.3cm} 7x+9y \leq 63, \hspace{0.3cm} x \leq 6, \hspace{0.3cm} y \leq 5, \hspace{0.3cm}x \geq 0, y \geq 0$ v) $2x+3y \leq 35 , \hspace{0.3cm} y \geq 3, \hspace{0.3cm} \hspace{0.3cm}x \geq 2, x \geq 0, y \geq 0$  $\\$

Question 2: Show that the solution set of the following linear inequations is empty set:

i) $x-2y \geq 0 , \hspace{0.3cm} 2x-y \leq -2, \hspace{0.3cm}x \geq 0, y \geq 0$

ii) $x+2y \leq 3 , \hspace{0.3cm} 3x+4y \geq 12, \hspace{0.3cm} y \geq 1, \hspace{0.3cm} x \geq 0, y \geq 0$

i) $x-2y \geq 0 , \hspace{0.3cm} 2x-y \leq -2, \hspace{0.3cm}x \geq 0, y \geq 0$ There is no common areas between the inequations. Hence there is no solution or the solution is null set.

ii) $x+2y \leq 3 , \hspace{0.3cm} 3x+4y \geq 12, \hspace{0.3cm} y \geq 1, \hspace{0.3cm} x \geq 0, y \geq 0$ There is no common areas between the inequations. Hence there is no solution or the solution is null set. $\\$

Question 3: Find the linear inequations for which the shaded area in the figure below is the solution set. Draw the diagram of the solution set of the linear in-equations. Answer:

The inequations are $4x+6y \leq 24, \ \ \ -3x+2y \leq 3 , \ \ \ x - 2y \leq 2, \ \ \ 2x+ 3y \geq 6$  $\\$

Question 4: Find the linear inequations for which the solution set is the shaded region given in the figure below. Answer:

The inequations are $x+y \leq 4, \ \ \ y \leq 3 , \ \ \ x \leq 3, \ \ \ 6x+2y \geq 8 \ \ \ x \geq 0, \ \ \ y\geq 0$  $\\$

Question 5: Show that the solutions set of the following linear inequations is an unbound set: $x+y \geq 9 , \hspace{0.3cm} 3x+y \geq 12, \hspace{0.3cm}x \geq 0, y \geq 0$

Solutions set of the following linear inequations is an unbound set.  $\\$

Question 6: Solve the following systems of inequations graphically:

i) $2x+y \geq 8, \hspace{0.3cm} x + 2y \geq 8, \hspace{0.3cm} x + y \leq 6$

ii) $12x+12y \leq 840, \hspace{0.3cm} 3x + 6y \leq 300, \hspace{0.3cm} 8x+4y \leq 480, \hspace{0.3cm} x \geq 0, y \geq 0$

iii) $x+2y \leq 40, \hspace{0.3cm} 3x + y \geq 30, \hspace{0.3cm} 4x+3y \geq 60, \hspace{0.3cm} x \geq 0, y \geq 0$

iv) $5x+y \geq 10, \hspace{0.3cm} 2x + 2y \geq 12, \hspace{0.3cm} x+4y \geq 12, \hspace{0.3cm} x \geq 0, y \geq 0$

i) $2x+y \geq 8, \hspace{0.3cm} x + 2y \geq 8, \hspace{0.3cm} x + y \leq 6$ ii) $12x+12y \leq 840, \hspace{0.3cm} 3x + 6y \leq 300, \hspace{0.3cm} 8x+4y \leq 480, \hspace{0.3cm} x \geq 0, y \geq 0$ iii) $x+2y \leq 40, \hspace{0.3cm} 3x + y \geq 30, \hspace{0.3cm} 4x+3y \geq 60, \hspace{0.3cm} x \geq 0, y \geq 0$ iv) $5x+y \geq 10, \hspace{0.3cm} 2x + 2y \geq 12, \hspace{0.3cm} x+4y \geq 12, \hspace{0.3cm} x \geq 0, y \geq 0$  $\\$

Question 7: Show that the following system of linear inequations have some solution: $x+2y \leq 3, \hspace{0.3cm} 3x + 4y \geq 12, \hspace{0.3cm} x \geq 0, y \geq 1$

Answer: There is no solution set for the given inequalities. $\\$

Question 8:  Show that the solution set of the following system of linear inequations is an unbounded region $2x+y \geq 8, \hspace{0.3cm} x + 2y \geq 10, \hspace{0.3cm} x \geq 0, y \geq 0$

We see that the solution set is the unbounded region.  $\\$