Chapter 10.9, Problem 24E

To determine

Therefore, the coordinates the vertex is [(4,4)]

Answer to Problem 24E

The solution region is an unbounded region.

Therefore, the coordinates the vertex is [(4,4)]

Explanation of Solution

Given:

  $\{\begin{array}{l}2x+3y>12\hfill \\ 3x-y<21\hfill \end{array}$

Calculation:

Consider the system of inequalities

  $\{\begin{array}{l}x-y>0\hfill \\ 4+y\le 2x\hfill \end{array}$

Sketch the graph of the solution of the system of inequalities.

Step 1: Graph the lines that corresponds to each inequality $x-y>0$ and $4+y\le 2x$

To graph the inequality $x-y>0$ simply graph the boundary line $x-y=0,$ which is a line.

Clearly, the points on the line itself do not satisfy the inequality because, the inequality contains > symbol so, graph the boundary line with a dashed curve.

To graph the inequality $4+y\le 2x$ simply graph the boundary line $4+y=2x,$ which is also a line.

Clearly, the points on the line itself satisfy the inequality because, the inequality contains $\le$ symbol so, graph the boundary line with a solid curve.

Step 2: To determine whether which region satisfies the inequalities, use the test point (0,0)

Substitute the coordinates of the point (0,0) in the inequalities and check whether the result satisfies the inequality or not.

The computations are tabulated as shown below:

Inequality	Test point (0,0)	Conclusion
$x-y>0$	$\begin{array}{l}0-0>0\\ 0>0\end{array}$	Not a part of the graph
$4+y\le 2x$	$\begin{array}{l}4+0\le 2\left(0\right)\\ 4\cancel{\le }0\end{array}$	Not a part of the graph

From the above table, observe that

1. The point (0,0) does not satisfy the inequality $x-y>0$ so, shade the region on or below

the dashed line that does not contain the point (0,0)

2. The point (0,0) does not satisfy the inequality $4+y\le 2x$ so, shade the region on or below the solid line that does not contain the point (0,0)

The graph of the solution of the system of inequalities is shown below:

  

Determine the vertices:

Solve the equations of the lines:

From the system

  $\{\begin{array}{ll}x-y=0\hfill & \text{ Equation }1\hfill \\ 4+y=2x\hfill & \text{ Equation }2\hfill \end{array}$

Solve the equation $1x-y=0$ for $x$

  $x-y+y=0+y$ Add $y$ on both sides

  $x=y$

Substitute $x=y$ in the equation $2,$ then

  $4+y=2x$

  $4+y=2y$ Substitute $y$ for $x$

  $4=y$

Subtract $y$ for both sides

Clearly, $x=4$ because $x=y$

Therefore, the coordinates the vertex is [(4,4)]

The solution set (intersection) and the vertex is shown as below:

  

From the graph observe that the solution region is an unbounded region.

Conclusion:

Therefore, the solution region is an unbounded region.

Therefore, the coordinates the vertex is [(4,4)]