Chapter 10.9, Problem 25E

To determine

To find: the coordinates of all the vertices and determine whether the solution set bounds or not.

Answer to Problem 25E

The vertices are $[(0,0),(4,0),(0,5)\text{ and }(2,4)]$

The solution set is bounded

Explanation of Solution

Given:

  $y\le -2x+8$

  $y\le -\frac{1}{2}x+5$

Calculation:

We obtain the equations from the given inequalities as

  $y=-2x+8$

  $y=-\frac{1}{2}x+5$

From these two equations, solve for $x$ we have

  $-2x+8=-\frac{1}{2}x+5$

  $-2x+\frac{1}{2}x=5-8$

  $-\frac{3}{2}x=-3$

  $x=2$

Substitute $x=2$ into the first equation.

  $y=-2x+8$

  $y=-2(2)+8$

  $y=-4+8$

  $y=4$

Therefore, the intersection point of the two equations is (2,4) .

We graph the lines given by the equations that correspond to each inequality. To determine the graphs of the linear inequalities, we need to check our test point. Here, we take (0,0) as test point.

Inequality	Test point (0,0)	Conclusion
$y\le -2x+8$	$0\le 8$	Satisfied inequality
$y\le -\frac{1}{2}x+5$	$0\le 5$	Satisfied inequality

Since (0,0) is below the lines $y=-2x+8$ and $y=-\frac{1}{2}x+5,$ our check shows that the region on or below the lines must satisfy the inequalities. The inequalities $x\ge 0$ and $y\ge 0$ shows that $x$ and $y$ are nonnegative.

Therefore, graph with shaded region is

  

Vertices:- The co-ordinate of one vertex is the point of intersection of the given lines, which is clearly (2,4). The origin (0,0) is also a vertex. The other two vertices are at the $x$ and $y$ intercept of the corresponding lines: (4,0) and (0,5) .

Therefore, the vertices are $[(0,0),(4,0),(0,5)\text{ and }(2,4)]$

From the figure it is clear that the shaded region is bounded. Therefore, the solution set is bounded

Conclusion:

Therefore, the vertices are $[(0,0),(4,0),(0,5)\text{ and }(2,4)]$ and the solution set is bounded