In the following figure,the feasible region (shaded) for a $LPP$ is shown. Determine the maximum and minimum value of $Z=x+2y$.

(A) From the figure,we have a bounded region with corner points $P\left(\frac{3}{13}, \frac{24}{13}\right)$,$Q\left(\frac{3}{2}, \frac{15}{4}\right)$,$R\left(\frac{7}{2}, \frac{3}{4}\right)$,and $S\left(\frac{18}{7}, \frac{2}{7}\right)$.
We evaluate the objective function $Z = x + 2y$ at each corner point:

Corner Point $(x, y)$	Value of $Z = x + 2y$
$P\left(\frac{3}{13}, \frac{24}{13}\right)$	$\frac{3}{13} + 2\left(\frac{24}{13}\right) = \frac{3+48}{13} = \frac{51}{13} \approx 3.92$
$Q\left(\frac{3}{2}, \frac{15}{4}\right)$	$\frac{3}{2} + 2\left(\frac{15}{4}\right) = \frac{6+30}{4} = \frac{36}{4} = 9$
$R\left(\frac{7}{2}, \frac{3}{4}\right)$	$\frac{7}{2} + 2\left(\frac{3}{4}\right) = \frac{14+6}{4} = \frac{20}{4} = 5$
$S\left(\frac{18}{7}, \frac{2}{7}\right)$	$\frac{18}{7} + 2\left(\frac{2}{7}\right) = \frac{18+4}{7} = \frac{22}{7} \approx 3.14$

Comparing the values of $Z$ at all corner points,the maximum value is $9$ at point $Q$ and the minimum value is $\frac{22}{7}$ at point $S$.