Solution:

The given inequalities are \[2x-y>1\], \[x-2y<-1\]

\[2x-y>1\]……………… (i)

Let us put value of \[x=0\] and \[y=0\]in equation one by one, we get

\[y=-1\]and \[x=0.5\]

We got the points as \[(0,-1)\]and \[(0.5,0)\]

Now check for the origin\[(0,0)\]

We got \[0>1\], which is false

Therefore, the origin does not lie in the solution region.

Hence the required region would be on the right of the line`s graph.

\[x-2y<-1\]………… (ii)

Let us put value of \[x=0\] and \[y=0\]in equation one by one, we get

\[y=0.5\]and \[x=-1\]

We got the points as \[(0,0.5)\]and \[(-1,0)\]

Now check for the origin \[(0,0)\]

We got \[0<-1\]which is false

Therefore, the origin does not lie in the solution area

Hence, the required area would be on the left side of the line`s graph.

In the graph below, the shaded area is the required solution of the given inequalities.