Chapter 6.6, Problem 32PPE

a.

To determine

To determine the graph of system inequality.

Explanation of Solution

Given:

  $x\ge 2$

  $y\ge -3$

  $x+y\le 4$

To draw the graph of $x\ge 2$ , $y\ge -3$

  $x+y\le 4.$

The representation will be,

Graph:

  

Thus, this is the graphical representation of $x\ge 2$ , $x+y\le 4$ and $y\ge -3.$

Interpretation:

Therefore, this is the graph of system inequality $x\ge 2$ , $x+y\le 4$ and $y\ge -3$ .

b.

To determine

To determine the shape of the solution region.

Answer to Problem 32PPE

Triangle

Explanation of Solution

Given:

  $x\ge 2$

  $y\ge -3$

  $x+y\le 4$

From the graphical representation of $x\ge 2$ , $y\ge -3$ and $x+y\le 4$ ,

The representation from the previous section is,

Graph:

  

And the shape formed by the overlapping shaded region is triangle.

Interpretation:

Therefore, the shape of the solution region is triangle.

c.

To determine

To determine the vertices of the solution region.

Answer to Problem 32PPE

  $\left(2,2\right),\left(2,-3\right)\text{ and }\left(7,-3\right)$

Explanation of Solution

Given:

  $x\ge 2$

  $y\ge -3$

  $x+y\le 4$

From the graphical representation of $x\ge 2$ , $y\ge -3$ and $x+y\le 4$ ,

The representation from the previous section is,

Graph:

  

The vertices of the triangle are the point where the three solution region interact or intersect it other or overlap each other.

The vertices are $\left(2,2\right),\left(2,-3\right)\text{ and }\left(7,-3\right)$as they are represented on the graph.

Interpretation:

Therefore, the vertices of the solution region is $\left(2,2\right),\left(2,-3\right)\text{ and }\left(7,-3\right)$ .

d.

To determine

To determine the area of the solution region.

Answer to Problem 32PPE

  $\frac{25}{2}$ Square unit or $12.5$

Explanation of Solution

Given:

  $x\ge 2$

  $y\ge -3$

  $x+y\le 4$

Formula used:

Area of triangle $=\frac{1}{2}\times \text{base}\times \text{height}$

Calculation:

From the graphical representation of $x\ge 2$ , $y\ge -3$ and $x+y\le 4$ ,

The representation from the previous section is,

Graph:

  

  $\Rightarrow$ Now from the vertices the value of base $=(7-2)=5$ [Base is along $x$ axis]

  $\Rightarrow$ Now from the vertices the value of height $=(2-(-3))=(2+3)=5$ [Height is along $y$ axis]

  $\Rightarrow$ Area of triangle $=\frac{1}{2}\times \text{base}\times \text{height}$

Putting values of base and height

  $\Rightarrow$ Area of triangle $=\frac{1}{2}\times 5\times 5$

  $\Rightarrow$ Area of triangle $=\frac{1}{2}\times 25$

  $\Rightarrow$ Area of triangle $=\frac{25}{2}$ Square unit or $12.5$ .

Interpretation:

Therefore, the area of the solution region is $\frac{25}{2}$ Square unit or $12.5$ .