Chapter 7.5, Problem 54E

To determine

To write: The system of inequalities that describes the region shown in the given graph.

Answer to Problem 54E

The system of inequalities that describes the region in the graph is

  $\{\begin{array}{c}x\ge 0\\ y\ge 0\\ y\ge -x+4\\ y\ge -\frac{1}{4}x+2\end{array}$ .

Explanation of Solution

Given information:

Thegraph of a system of inequalities is given below:

  

From the graph, it is evident that the region falls in the first quadrant. Therefore, $x\ge 0\text{and}y\ge 0.$

Now, observing the graph, one line has the x -intercept at $(4,0)$ and y -intercept at $(0,4)$ . So the equation of the line would be

  $\begin{array}{l}slope=\frac{4-0}{0-4}=-1\\ y=-x+4\end{array}$

Since the region falls above the line, the inequality corresponding to the line would be $y\ge -x+4$ .

Similarly, other line has the x -intercept at $(8,0)$ and y-intercept at $(0,2)$ . So the equation of the line would be

  $\begin{array}{l}slope=\frac{2-0}{0-8}=-\frac{1}{4}\\ y=-\frac{1}{4}x+2\end{array}$

Since the region falls above the line, the inequality corresponding to the line would be $y\ge -\frac{1}{4}x+2$ .

Conclusion:

Therefore the system of inequalities that describes the region in the graph is

  $\{\begin{array}{c}x\ge 0\\ y\ge 0\\ y\ge -x+4\\ y\ge -\frac{1}{4}x+2\end{array}$