Exercise 5 Page 160

We will first graph the system, then name the vertices of the overlapping region, which can be also called feasible. Note that two of the given inequalities contain only the $x\text{-}$variable and two $-$ only the $y\text{-}$variable. We will graph both cases, for $x\text{-}$ and for $y\text{-}$values, separately. Next, we will combine the graphs and identify the overlapping region. Let's start!

Inequalities I and III

First, let's graph the inequalities which contain only the $x\text{-}$variable. Note that we can form a compound inequality from these two cases.

$$\begin{array}{rl}\text{Case I: }& x\le 5\\ \text{Case II: }& 0\le x\\ \text{Compound Inequality: }& 0\le x\le 5\end{array}$$

This compound inequality describes all values of $x$ that are greater than or equal to $0$ and less than or equal to $5.$ This means that all coordinate pairs with an $x\text{-}$coordinate that is greater than or equal to $0$ and less than or equal to $5$ will be in the solution set. Note that both inequalities are non-strict, so the boundary lines will be solid.

Inequalities II and IV

For inequalities containing only the $y\text{-}$variable we will follow a similar process. Let's form the compound inequality.

$$\begin{array}{rl}\text{Case I: }& y\le 4\\ \text{Case II: }& 0\le y\\ \text{Compound Inequality: }& 0\le y\le 4\end{array}$$

This compound inequality describes all values of $y$ that are greater than or equal to $0$ and less than or equal to $4.$ This means that all coordinate pairs with a $y\text{-}$coordinate that is greater than or equal to $0$ and less than or equal to $4$ will be in the solution set. Note that both inequalities are non-strict, so the boundary lines will be solid.

Combining the Inequality Graphs

Let's draw the graphs of the inequalities on the same coordinate plane.

Now that we can see the overlapping region, let's highlight the vertices.

Looking at the graph, we can see that the line $y=0$ intersects the lines $x=0$ and $x=5$ at the points $(0,0)$ and $(5,0),$ respectively. These are two of the vertices.

$$\begin{array}{c}(0,0)\text{ and }(5,0)\end{array}$$

Similarly, we can see that the line $y=4$ intersects the lines $x=0$ and $x=5$ at the points $(0,4)$ and $(5,4),$ respectively. We have identified two remaining vertices.

$$\begin{array}{c}(0,4)\text{ and }(5,4)\end{array}$$