Show that the solution set of the following linear constraints is empty: $x-2 y \geq 0,2 x-y \leq-2, x \geq 0$ and $y \geq 0$

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Show that the solution set of the following linear constraints is empty: $x-2 y \geq 0,2 x-y \leq-2, x \geq 0$ and $y \geq 0$

Consider the inequation $x-2 y \geq 0$
$\begin{array}{l} \Rightarrow x \geq 2 y \\ \Rightarrow y \leq \frac{x}{2} \end{array}$
consider the equation $y=\frac{x}{2}$ . This equation’s graph is a straight line passing through origin.
Now consider the inequality $y \leq \frac{x}{2}$
Here we need the y value less than or equal to $\frac{x}{2}$
$\Rightarrow$ the required region is below origin.
Therefore the graph of the inequation $y \leq \frac{x}{2}$ is fig. $10 a$

Consider the inequation $2 x-y \leq-2:$

\Rightarrow y \geq 2 x+2

Consider the equation $y=2 x+2$
Finding points on the coordinate axes:
If $x=0$ , the $y$ value is 2 i.e, $y=2$
$\Rightarrow$ the point on $Y$ axis is $A(0,2)$
If $y=0,0=2 x+2$

\Rightarrow \mathrm{x}=-1

The point on $X$ axis is $B(-1,0)$
Now consider the inequality $y \geq 2 x+2$
Here we need the y value greater than or equal to $2 \mathrm{x}+2$
$\Rightarrow$ the required region is above point $A$ .
Therefore the graph of the inequation $y \geq 2 x+2$ is fig. $10 b$