Chapter 7, Problem 57RE

Solution

To graph: The given inequality $2x-y\le 1$ .

The required graph is show as following,

  

Given Information:

The inequality is defined as,

  $2x-y\le 1$

Explanation:

Consider the given inequality,

  $2x-y\le 1$

The boundary is included if  $\le$ or  $\ge$  is used and is excluded if  $<$  or $>$  is used. As $>$  was used, then the boundary is excluded from the graph.

So, the boundaries are included.

Replacing the inequality symbol with the  $=$  symbol, the equation for the boundary is  $2x-y=1$ , which is a slant line.

Now, find the intercepts of the equation by substituting 0 for x and 0 for y one by one.

  $\begin{array}{c}2x-\left(0\right)=1\\ x=\frac{1}{2}\end{array}$

And,

  $\begin{array}{c}2\left(0\right)-y=1\\ y=-1\end{array}$

Therefore, the intercept is $\left(\frac{1}{2},0\right)\text{ and }\left(0,-1\right)$ .

To determine the half-plane to be shaded, use a test point not on the boundary, say  $\left(0,0\right)$ .

  $\begin{array}{c}2\left(0\right)-0\le 1\\ 0\le 1\end{array}$

The statement is true so shade the half-plane where  $\left(0,0\right)$ is located. The graph will be:

Draw the given inequality.

  

Hence, the above graph is the required graph.