Chapter 7.4, Problem 9E

Solution

To Graph: The given inequality by using the hand sketch and state the boundary region.

  $2x+5y\le 7$

The boundary $2x+5y=7$ is included in the graph, and the graph of the inequality is shown as below,

  

Given Information:

The inequality is defined as,

  $2x+5y\le 7$

Explanation:

Consider the given inequality,

  $2x+5y\le 7$

Replacing the inequality symbol with the = symbol, the equation for the boundary is  $2x+5y=7$ , which is a slanted line.

Now, find the intercept of the line. Substitute 0 for x and then 0 for y in the line one by one.

  $\begin{array}{c}2x+5\left(0\right)=7\\ 2x=7\\ x=3.5\end{array}$

And,

  $\begin{array}{c}2\left(0\right)+5y=7\\ 5y=7\\ y=0.14\end{array}$

Therefore, the intercept is $\left(3.5,0\right)\text{ and }\left(0,0.14\right)$ .

The boundary point is included if the inequality containing $\le \text{ or }\ge$ symbols and the boundary point is excluded if the symbols $<\text{ or }>$  is used.

As the less then equal to symbol was used, then the boundary is included in the graph and drawing it by using solid line. To determine the half-plane to be shaded, use a test point not on the boundary, say  $\left(0,0\right)$ . Then,

  $\begin{array}{c}2\left(0\right)+5\left(0\right)\le 7\\ 0\le 7\end{array}$

The statement is true so shade the half-plane where  $\left(0,0\right)$

 is located. The graph will be:

  

Hence, the required graph is shown above and the boundary $2x+5y=7$ is included in the graph.