Chapter 11, Problem 26CT

To determine

To graph: The system of inequalities, $\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+2y\ge 8\\ 2x-3y\ge 2\end{array}$ and state whether the graph is bounded or unbounded and label all corner points.

Explanation of Solution

Given information:

The system of inequalities, $\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+2y\ge 8\\ 2x-3y\ge 2\end{array}$.

Graph:

An inequality $x\ge 0$ and $y\ge 0$ indicates that the graph lies in the first quadrant.

Now for the remaining inequalities, $x+2y\ge 8$ and $2x-3y\ge 2$.

To plot the graph of an inequality, replace inequality sign by equal sign and find the boundary curve.

$\Rightarrow x+2y=8$ and $2x-3y=2$

If inequality is of type $\le ,\ge$ then boundary curve would be solid, if inequality is of type $<,>$ then boundary curve would be dashed curve.

Here in $x\ge 0,y\ge 0,x+2y\ge 8$ and $2x-3y\ge 2$ inequality sign is $\ge$ so boundary curve would be solid.

An equation, $x+2y=8$ represents an equation of line.

Find some ordered pairs of $x+2y=8$

For that choose $x$ values and calculate corresponding $y$ values. To draw a line at least two points to be known.

Choose $x=0$ and plug in $x+2y=8$ gives $2y=8\Rightarrow y=4$

Choose $x=2$ and plug in $x+2y=8$ gives $2+2y=8\Rightarrow 2y=6\Rightarrow y=3$

So, the two points on a line $x+2y=8$ are $\left(0,4\right),\left(2,3\right)$.

Now an equation, $2x-3y=2$ represents an equation of line.

Find some ordered pairs of $2x-3y=2$

Choose $x=1$ and plug in $2x-3y=2$ gives $-3y=0\Rightarrow y=0$

Choose $x=4$ and plug in $2x-3y=2$ gives $-3y=-6\Rightarrow y=2$

So the two points on a line $2x-3y=2$ are $\left(1,0\right),\left(4,2\right)$.

Nowby connecting points on each line to draw a linethat represents a graph below:

Now, to plot the graph of system of inequality $\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+2y\ge 8\\ 2x-3y\ge 2\end{array}$

Choose a test point and substitute in inequality, if test point satisfies the inequality shade the region which includes test point otherwise shade the region which does not include test point.

For $x+2y\ge 8$, consider $\left(0,0\right)$ as test point.

Substitute $x=0,y=0$ in $x+2y\ge 8$

$\Rightarrow 0\ge 8$ False, test point does not satisfy the inequality. So, shade the region which does not includes test point $\left(0,0\right)$.

For $2x-3y\ge 2$, consider $\left(0,0\right)$ as test point.

Substitute $x=0,y=0$ in $2x-3y\ge 2$

$\Rightarrow 0\ge 2$ False, test point does not satisfy the inequality. So, shade the region which does not includes test point $\left(0,0\right)$.

Therefore, The graph of system of inequality $\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+2y\ge 8\\ 2x-3y\ge 2\end{array}$ is:

Thus the intersection of the shaded region gives the resulting graph with corner points are labeled is shown below,

Interpretation:

The graph represents a system of inequalities, $\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+2y\ge 8\\ 2x-3y\ge 2\end{array}$ is unbounded and the corner points are $\left(4,2\right),\left(8,0\right)$.