Chapter 7.5, Problem 31E

To determine

Sketch the graph of the solution set of system of inequalities and label the vertices of the region.

Answer to Problem 31E

The graph for the first inequality is the half-plane to the left of the line (1)

The graph for the second inequality is the half-plane lying to the right of the line (2)

The graph for the third inequality is the half-plane lying above the line (3)

Explanation of Solution

Given:

The system of linear inequalities is:

  $\begin{array}{r}\hfill x+y\le 1\\ \hfill -x+y\le 1\\ \hfill y\ge 0\end{array}$

Now, find the points of each pair of equations.

  $\begin{array}{l}x+y=\mathrm{1..........}(1)\hfill \\ -x+y=\mathrm{1........}(2)\hfill \\ y=\mathrm{0................}(3)\hfill \end{array}$

First, solve equations (1) and (2),

  $\begin{array}{r}\hfill x+y=1\\ \hfill -x+y=1\end{array}$

On adding these two equations,

  $\begin{array}{l}2y=2\\ y=1\end{array}$

Replace the value of $y$ in equation (1),

  $\begin{array}{l}x+1=1\\ x=0\end{array}$

Therefore, the point of intersection is (0, 1).

Now, solve equations (1) and (3)

  $\begin{array}{l}x+y=1\\ y=0\end{array}$

Thus,

  $\begin{array}{cc}-x+0& =1\\ x& =-1\end{array}$

Therefore, the point of intersection is (-1, 0)

The graph for the first inequality is the half-plane to the left of the line (1).

The graph for the second inequality is the half-plane lying to the right of the line (2)

The graph for the third inequality is the half-plane lying above the line (3)

Therefore, the graph with the solution set of the system of inequalities is shown below: