Chapter 10.9, Problem 51E

To determine

The system of inequalities for number of fiction and nonfiction books. the company can produce and sketch the graph for solution set.

Answer to Problem 51E

The system of inequalities that describes the number of fiction and nonfiction books the company produces are $\{\begin{array}{l}x+y\le 100\\ 20\le y\\ x\ge y\\ x\ge 0\\ y\ge 0\end{array}$, and the graph is shown in Figure (1).

Explanation of Solution

Given:

Let the number of fiction books the company produces be x

Let the number of nonfiction books the company produces be y

The total number of books produced every year is not greater than $100$, and represented as inequality,

$x+y\le 100$

At least twenty of the books are nonfiction, and represented as inequality,

$y\ge 20$

The company publishes at least as much fiction as nonfiction.

So, $x\ge y$.

The number of books cannot be negative.

$x\ge 0$, $y\ge 0$

Take the first inequality $x+y\le 100$.

To plot the graph change the sign of inequality as $x+y=100$.

Substitute 0 for x in $x+y=100$

$y=100$

Substitute 0 for y in $x+y=100$

$x=100$

Take the second inequality $y\ge 20$

To plot the graph change the sign of inequality as $y=20$

$y=20$ is a straight line parallel to x-axis.

Now, $x\ge 0,y\ge 0$ represents first quadrant.

The point of intersection between the first two inequalities is calculated by equating equation of one inequality into equation of other.

Substitute $20$ for y in $x+y=100$.

$\begin{array}{c}x+20=100\\ x=80\end{array}$

The point of intersection between $y=20$ and $x+y=100$ is $\left(80,20\right)$.

The point of intersection between $y=20$ and $x=y$ is $\left(20,20\right)$

The point of intersection between $x+y=100$ and $x=y$ is calculated by equating one equation into other.

Substitute y for x in $x+y=100$

$\begin{array}{c}2y=100\\ y=50\end{array}$

Substitute $50$ for y in $x+y=100$

$x=50$

The point of intersection between $x+y=100$ and $x=y$ is $\left(50,50\right)$.

Figure (1)

Thus, the system of inequalities that describes the number of fiction and non-fiction books the company produces are $\{\begin{array}{l}x+y\le 100\\ 20\le y\\ x\ge y\\ x\ge 0\\ y\ge 0\end{array}$, and the graph is shown in Figure (1).