Chapter 5.7, Problem 34E

(a)

To determine

To write: a system of linear inequalities represented by the shaded triangle.

Answer to Problem 34E

The system of inequalities is $y\ge 2x+1,y\ge -3$ and $y\le -2x+9$ .

Explanation of Solution

Given information:

The following points are the vertices of a shaded rectangle

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

Calculation:

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 $\left(-2,-3\right)$ and $\left(2,5\right)$ is

  $\begin{array}{l}y\ge \frac{5+3}{2+2}\left(x+2\right)-3\text{Slope}\text{point}\text{formula}\text{for}\text{line}\text{equation}\\ y\ge 2x+1\text{Simplify}\end{array}$

The inequality joining two points $\left(6,3\right)$ and $\left(2,5\right)$ is

  $\begin{array}{l}y\le \frac{5+3}{2-6}\left(x-6\right)-3\text{Slope}\text{point}\text{formula}\text{for}\text{line}\text{equation}\\ y\le -2x+9\text{Simplify}\end{array}$

Therefore, the system of inequalities is $y\ge 2x+1,y\ge -3$ and $y\le -2x+9$ .

(b)

To determine

To find: the area of the triangle.

Answer to Problem 34E

The area of the triangle is $32$ square units.

Explanation of Solution

Calculation:

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

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

Formula for area of a triangle is $\frac{1}{2}bh$ .

The area of the triangle is $\frac{1}{2}\times 8\times 8=32$ square units  .

Therefore, the area of the triangle is 32 square units.