Problems 43–52, graph each system of linear inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points

$52.\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+2y\ge 1\\ x+2y\le 10\\ x+y\ge 2\\ x+y\le 8\end{array}\right.$

Consider the first inequality in the system of inequalities is $x\ge 0$& $y\ge 0$.

$x+2y\ge 1$equation of the equality consider by replacing with equality $x+2y=1$

For determining the shaded portion we use to test a point with $(0,0)$

$$\begin{gathered} 0+0\ge 1 \\ 0\ge 1 \end{gathered}$$

$x+2y\le 10$,the equation of line is replacing by inequality with equality$x+2y=10$

By substituting by test point $(0,0)$

$$\begin{gathered} 0+0\le 10 \\ 0\le 10 \end{gathered}$$

The point $(0,0)$contains the shaded portion in the equation $x+2y\le 10$

$x+y\ge 2$, the equation of line replacing with equality $x+y=2$

By using the testing point $(0,0)$

$$\begin{gathered} 0+0\ge 2 \\ 0\ge 2 \end{gathered}$$

The point$(0,0)$does not includes with the inequality$x+y\ge 2$

$x+y\le 8$,the equation of line replace with equality $x+y=8$

By using test point$(0,0)$

$0+0\le 8$

role="math" localid="1646922749498" $0\le 8$

The point $(0,0)$includes the inequality$x+y\le 8$

The shaded portion is the bounded region: