Chapter 10, Problem 14T

(a)

To determine

To sketch: The graph of the system of inequalities and label the vertices with their coordinates.

Explanation of Solution

The system of inequalities is,

$\{\begin{array}{l}2x+y\le 8\\ x-y\ge -2\\ x+2y\ge 4\end{array}$

Check the inequality by a test point $\left(0,0\right)$.

Inequality	Test point (0,0)	Conclusion
$2x+y\le 8$	$2\left(0\right)+0\le 8$	Satisfies inequality
$x-y\ge -2$	$0-0\ge -2$	Satisfies inequality
$x+2y\ge 4$	$0+2\times 0\ge 4$	Not satisfy inequality

Draw the graph of the inequality.

Figure (1)

Since $\left(0,0\right)$ is below the lines $2x+y\le 8$ and $x-y\ge -2$, our check shows that the region on or below the line must satisfy the inequality.

Since $\left(0,0\right)$ is below the lines $x+2y\ge 4$ and it does not satisfy the inequality thus, the solution lies above the line $x+2y\ge 4$.

The solution of the system of inequalities is the intersection of the graphs.

From Figure (1), the solution of the system of inequality are $\left(0,2\right)$, $\left(4,0\right)$ and $\left(2,4\right)$.

Thus, the vertices of the system of inequalities are $\left(0,2\right)$, $\left(4,0\right)$ and $\left(2,4\right)$.

(b)

To determine

To sketch: The graph of the system of inequalities and label the vertices with their coordinates.

Explanation of Solution

The system of inequalities is,

$\{\begin{array}{l}{x}^2+y\le 5\\ y\ge 2x+5\end{array}$

Check the inequality by a test point $\left(0,0\right)$.

Inequality	Test point (0,0)	Conclusion
$x^2+y\le 5$	$0+0\le 5$	Satisfies inequality
$y\ge 2x+5$	$0\ge 2\left(0\right)+5$	Not satisfy inequality

Draw the graph of the inequality.

Figure (2)

Since $\left(0,0\right)$ is below the curve $x^2+y\le 5$, our check shows that the region on or below the curve must satisfy the inequality.

Since $\left(0,0\right)$ is below the line $y\ge 2x+5$ and it does not satisfy the inequality thus, the solution lies above the line $x+2y\ge 4$.

The solution of the system of inequalities is the intersection of the graphs.

From Figure (2), the solution of the system of inequality are $\left(-2,1\right)$ and $\left(0,5\right)$.

Thus, the vertices of the system of inequalities are $\left(-2,1\right)$ and $\left(0,5\right)$.