Chapter 5.7, Problem 34E

(a)

To determine

To find: the system of linear inequalities represented by the shaded region.

Answer to Problem 34E

  $y\ge 2x+1,y\le -2x+9\text{and }y\ge -3.$

Explanation of Solution

Given:

Vertices of triangle:

  $\left(2,5\right),\left(6,-3\right),\left(-2,-3\right)$ .

Concept used:

The difference is that a solution to the inequality is not the drawn line but area of the coordinate plane that satisfy the inequality.

The common area is the solution of inequality.

The boundary line is dashed for $<\text{or >}$ and solid line for $\le \text{or }\ge$ .

Straight line equation:

  $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Calculation:

The vertices of a triangle are:

  $\left(2,5\right),\left(6,-3\right),\left(-2,-3\right)$ .

The vertices $\left(6,-3\right),\left(-2,-3\right)$ are having same $y$ coordinate and less than $y$ coordinate of the vertex $\left(2,5\right)$ . it means one boundary line of the rectangle is:

  $y\ge -3$ .

The inequality joining two points $\left(-2,-3\right)\text{ and }\left(5,2\right)$ is:

  $\begin{array}{l}y\ge \frac{5+3}{2+2}\left(x+2\right)-3.\\ y\ge 2x+1.\end{array}$

The inequality joining two points $\left(6,-3\right)\text{ and }\left(2,5\right)$ is:

  $\begin{array}{l}y\ge \frac{5+3}{2-6}\left(x-6\right)-3.\\ y\ge -2x+9.\end{array}$

Hence, the system of inequalities is

  $y\ge 2x+1,y\le -2x+9\text{and }y\ge -3.$

(b)

To determine

To find: the area of the triangle.

Answer to Problem 34E

The area of triangle is $32\text{sq units}\text{.}$

Explanation of Solution

Given:

Vertices of triangle:

  $\left(2,5\right),\left(6,-3\right),\left(-2,-3\right)$ .

Concept used:

The difference is that a solution to the inequality is not the drawn line but area of the coordinate plane that satisfy the inequality.

The common area is the solution of inequality.

The boundary line is dashed for $<\text{or >}$ and solid line for $\le \text{or }\ge$ .

Straight line equation:

  $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$ .

Area of triangle is:

  $\frac{1}{2}\times b\times h$ .

Calculation:

The length of the base of the triangle is difference between $x$ points of $-2\text{ and }6$ which is $8$ units.

The height of the triangle is difference between $y$ points of $\text{-3and }5$ which is $8$ units.

The are of triangle is $\frac{1}{2}\times 8\times 8=32\text{sq units}\text{.}$

Applying formula of triangle:

  $\frac{1}{2}\times b\times h$ .

Hence, the area of triangle is $32\text{sq units}\text{.}$