Chapter 7.5, Problem 58E

To determine

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

Answer to Problem 58E

The system of inequalities that describes the given triangle region is

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

Explanation of Solution

Given information:

Theregion of a system of inequalities is a triangle with vertices: $(-1,0),(1,0),(0,1)$ .

The rough graph of the triangle region enclosed with the given vertices is shown below:

  

The sides of a triangle are line segments, so the inequalities are linear inequalities.

To find the linear inequality corresponding to the points $(-1,0)\text{and}(1,0):$

  $\begin{array}{l}slope=\frac{0-0}{1-(-1)}=0\\ y=0\end{array}$

Since the triangle region is above the line $y=0$ , so the linear inequality would be $y\ge 0$ .

To find the linear inequality corresponding to the points $(-1,0)\text{and}(0,1):$

  $\begin{array}{l}slope=\frac{1-0}{0-(-1)}=1\\ y=1x+b(\because y=mx+b)\\ \Rightarrow 0=1(-1)+b(\text{At }(-1,0))\\ \Rightarrow b=1\\ \therefore y=x+1\end{array}$

Since the triangle region is right side of the line $y=x+1$ and it has the positive slope, so the linear inequality would be $y\le x+1$ .

Now, to find the linear inequality corresponding to the points $(0,1)\text{and }(1,0):$

  $\begin{array}{l}slope=\frac{0-1}{1-0}=-1\\ y=-1x+b(\because y=mx+b)\\ \Rightarrow 0=-1(1)+b(\text{At }(1,0))\\ \Rightarrow b=1\\ \therefore y=-x+1\end{array}$

Since the triangle region is left side of the line $y=-x+1$ and it has the negative slope, so the linear inequality would be $y\le -x+1$ .

Conclusion:

Therefore the system of inequalities that describes the triangle region is

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