Question

Consider the following statements
Statement I The solution set of the system of inequalities $5 x+4 y \leq 20, x \geq 1, y \geq 2$ is

Statement II The solution set of the inequalities $3 x+4 y \leq 60, x+3 y \leq 30, x \geq 0, y \geq 0$ 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: (B)

I. The given system of inequalites
$5 x+4 y \leq 20 $...(i)
$x \geq 1 $ ...(ii)
$y \geq 2$ ...(iii)
Step I Consider the inequations as strict equations i.e., $5 x+4 y=20$
$x=1, y=2$
Step II Find the points on the $X$-axis and $Y$-axis for $5 x+4 y=20$

Step III Plot the graph using the above tables.
(i) For $5 x+4 y=20$ use the above table.
(ii) Graph of $x=1$ will be line parallel to $Y$-axis, intersecting the $X$-axis at 1.
(iii) Graph of $y=2$ will be a line parallel to $X$-axis, intersecting the $Y$-axis at 2 .

Step IV Take a point $(0,0)$ and put it in the given inequations (i), (ii) and (iii),
$0+0 \leq 20 \text { (true) } $
$0 \geq 1 \text { (false) } $
$ 0 \geq 2 \text { (false) } $
Thus, common shaded region shows the solutions of the inequalities.
II. The given system of inequalities
$3 x+4 y \leq 60$...(i)
$x+3 y \leq 30$...(ii)
$x \geq 0 $...(iii)
$y \geq 0$...(iv)
Step I Consider the inequations as strict equations
$\text { i.e., }3 x+4 y =60, x+3 y=30$
$x =0, y=0$
Step II Find the points on the $X$-axis and $Y$-axis for

Step III Plot the graph using the above tables.
(i) For $3 x+4 y=60$ and $x+3 y=30$ use the above tables.
(ii) Graph of $x=0$, will be $Y$-axis.
(iii) Graph of $y=0$, will be $X$-axis
Step IV Take a point $(0,0)$ and put it in inequations (i) and (ii),
$0+0 \leq 60 \text { (true) }$
So, the shaded region will be towards origin. and
$0+0 \leq 30$(true)
So, the shaded region will be towards origin.

And point $(1,0)$ in inequation (iii),
$1 \geq 0$ (true)
So, the shaded region will be towards origin.
Point $(0,1)$ in inequation (iv),
$1 \geq 0$(true)
So, the shaded region will be towards origin.
Thus, common shaded region shows the solution of the inequalities.