Question

Consider the following statements
Statement I The solution set of the inequalities $x+2y\le 8,2x+y\le 8,x\ge 0,y\ge 0$ is

Statement II The solution set of the inequalities $x\ge 3$ and $y\ge 2$ is

Choose the correct option.

• A. Statement I is true
• B. Statement II is true
• C. Both are true
• D. Both are false

Answer 1

Answer: (A)

I. We draw the graphs of the lines $x+2y=8$ and $2x+y=8$. The inequality (1) and (2) represent the region below the two lines, including the point on the respective lines.

Since, $x\ge 0,y\ge 0$, every point in the shaded region in the first quadrant represent a solution of the given system of inequalities.
II. The given system of inequalities
$x\ge 3$ ...(i)
$y\ge 2$ ....(ii)
Step I Consider the inequations as strict equations i.e., $x=3,y=2$
Step II Plot the graph. The graph of $x=3$, (i.e., $y=0$ ) will be a line parallel to $Y$-axis, intersecting the $X$-axis at 3 . The graph of $y=2$ (i.e., $x=0$ ) will be a line parallel to $X$-axis, intersecting the $Y$-axis at 2 .Step III Take a point $(0,0)$ and put it in the given inequations (i) and (ii).
$\text{ i.e., }0\ge 3\text{ (false) }$
$0\ge 2\text{ (false) }$

$\therefore$ Both the graphs will be parallel and away from the origin.
Thus, common shaded region shows the solution of the inequalities.