Step 1:
The corner points of the feasible regions are $P(\large\frac{3}{13},\frac{24}{13})$,$Q(\large\frac{3}{2},\frac{15}{4})$,$R(\large\frac{7}{2},\frac{3}{4})$ and $S(\large\frac{18}{7},\frac{2}{7})$
The function with in this constraint is $Z=x+2y$
Now let us find the maximum and minimum values of the function ,with in the given constraint.
Step 2:
The values of the objective function at these points are :
For the point $P(\large\frac{3}{13},\frac{24}{13})$ the value of the objective function is $Z=x+2y\Rightarrow \large\frac{3}{13}$$+2\times \large\frac{24}{13}=\frac{51}{13}$
For the point $P(\large\frac{3}{2},\frac{15}{4})$ the value of the objective function is $Z=x+2y\Rightarrow \large\frac{3}{2}$$+2\times \large\frac{15}{4}=$$9$
For the point $R(\large\frac{7}{2},\frac{3}{4})$ the value of the objective function is $Z=x+2y\Rightarrow \large\frac{7}{2}$$+2\times \large\frac{3}{4}=$$5$
For the point $S(\large\frac{18}{7},\frac{2}{7})$ the value of the objective function is $Z=x+2y\Rightarrow \large\frac{18}{7}$$+2\times \large\frac{2}{7}=\frac{22}{7}$
Step 3:
The maximum value is $9$ at $Q(\large\frac{3}{2},\frac{15}{4})$
The minimum value is $\large\frac{22}{7}$ at $S(\large\frac{18}{7},\frac{2}{7})$