Chapter 5.7, Problem 33E

(a)

To determine

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

Answer to Problem 33E

The system of inequalities is $x\ge -1,x\le 6\text{ and }y\ge -3\text{ and }y\le 1$ .

Explanation of Solution

Given:

The vertices of a rectangle are:

  $\left(-1,1\right),\left(6,1\right),\left(6,-3\right),\left(-1,-3\right)$ .

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 $\le \text{or }\ge$ .

Calculation:

The vertices of a rectangle are:

  $\left(-1,1\right),\left(6,1\right),\left(6,-3\right),\left(-1,-3\right)$ .

Write the system of linear inequalities that represent the shaded rectangle.

The two vertical lines of the rectangle are passing through $x$ points of $-1\text{ and }6$ .

It means the inequalities for the vertical boundaries are:

  $x\ge -1\text{ and }x\le 6$ .

The two horizontal lines of the rectangle are passing through $y$ points of $-3$ and $1$ .

It means the in equalities for the horizontal boundaries are:

  $y\ge -3\text{ and }y\le 1$ .

Hence, the system of inequalities is $x\ge -1,x\le 6\text{ and }y\ge -3\text{ and }y\le 1$ .

(b)

To determine

To find: the area of the rectangle using the vertices of a shaded rectangle.

Answer to Problem 33E

The area of the rectangle is $28\text{sq units}\text{.}$

Explanation of Solution

Given:

The vertices of a rectangle are:

  $\left(-1,1\right),\left(6,1\right),\left(6,-3\right),\left(-1,-3\right)$ .

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 $\le \text{or }\ge$ .

Area of rectangle is:

  $l\times b$ .

Calculation:

The length of the rectangle along horizontal direction is difference between $x$ points of $-1\text{ and }6$ which is $7$ units.

The length of the rectangle along vertical direction is difference between $y$ points of $-3$ and $1$ which is $4$ units.

The area of the rectangle is $7\times 4=28\text{sq units}\text{.}$

Hence, the area of the rectangle is $28\text{sq units}\text{.}$

(c)

To determine

To find: The range of the function.

Answer to Problem 33E

Range is $\{g(x)|g(x)>0\}$ .

Explanation of Solution

Given: ${\text{g(x)=8}}^{\text{x}}$

Concept used:

The power of the any number must be real number.

Calculation:

Here range is $g\left(x\right)$

  $g\left(x\right)>0$

the value of the root should be greater than 0.

So, Range here must be greater than zero as shown in the graph.

Graph of the function ${\text{g(x)=8}}^{\text{x}}$

So, from the graph its clear that range $\text{g}\left(x\right)$ is always greater than 0.

The range here $\{g(x)|g(x)>0\}$ .

Hence range is $\{g(x)|g(x)>0\}$ .