Q. 46 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 46–48, graph each system of inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points.

$\left\{\begin{array}{l}-2x+y\le 2\\ x+y\ge 2\end{array}\right.$

Step-by-Step Solution

Answer

The graph of $\left\{\begin{array}{l}-2x+y\le 2\\ x+y\ge 2\end{array}\right.$is unbounded and the corner point is $\left(0,2\right)$ and the graph is

Step 1. Given data

The given system of inequality is  

$\left\{\begin{array}{l}-2x+y\le 2\\ x+y\ge 2\end{array}\right.$

Step 2. Properties of the graph of the inequality

Graph of $-2x+y=2$will be a line

now test the point $(0,0)$for inequality

$$\begin{gathered} -2x+y\le 2 \\ -2\left(0\right)+\left(0\right)\le 2 \\ 0\le 2 \end{gathered}$$

inequality satisfied so the origin will lie in the region of inequality

So shade the region from line$-2x+y=2$ toward the origin

Step 3. Properties of the graph of the inequality

Graph of $x+y=2$ will be lines

now test the point$(0,0)$ for inequality

$$\begin{gathered} x+y\ge 2 \\ 0+0\ge 2 \\ 0<2 \end{gathered}$$

inequality does not satisfy so the origin will not lie in the region of inequality $x+y\ge 2$

So shade the region above the line$x+y=2$ 

Step 4. graph of the system of inequality

Plot solid lines $-2x+y\le 2$ and shade the region from line toward the origin and plot the line $x+y\ge 2$and shade the region away from origin to plot the graph of $\left\{\begin{array}{l}-2x+y\le 2\\ x+y\ge 2\end{array}\right.$

Graph extended infinitely in positive x and y direction so the graph is unbounded

Boundry lines of both inequalities intersect at the point $(0,2)$ so corner point is$(0,2)$