Question

The region represented by the solution set of the system of inequalities $x-y\le 1,x+2y\le 8$, $2x+y\ge 2,x\ge 0,y\ge 0$ is

• A. bounded
• B. unbounded
• C. Does not exist
• D. None of these

Answer 1

Answer: (A)

Given inequalities are
$x-y\le 1$...(i)
$x+2y\le 8$...(ii)
$2x+y\ge 2$...(iii)
$x\ge 0$...(iv)
$y\ge 0$....(v)
The lines corresponding to (i), (ii) and (iii) are
$x-y=1$...(vi)
$x+2y=8$...(vii)
$2x+y=2$....(viii)

Draw the graph of line (vi), (vii) and (viii).
$\because 0-0\le 1$ i.e., $0\le 1$, which is true.

Therefore, inequality (i) represent the half plane made by the line (vi), which contains origin.
Now, $0+2(0)\le 8$ i.e., $0\le 8$, which is true. So, inequality (ii) represent the half plane made by line (vii) which contain origin.
Again, $2(0)+0\ge 2$ i.e., $0\ge 2$, which is not true.
$\therefore$ Inequality (iii) represent the half plane made by the line (viii), which does not contain origin.
Also, $x\ge 0,y\ge 0$ represent the region in I quadrant.
The region common to (i), (ii), (iii), (iv) and (v) is the shaded region which is bounded.