Problem 45 - Algebra and Trigonometry

Question

Publishing Books A publishing company publishes a total of no more than 100 books every year. At least 20 of these are nonfiction, but the company always publishes at least as much fiction as nonfiction. Find a system of inequalities that describes the possible numbers of fiction and nonfiction books that the company can produce each year consistent with these policies. Graph the solution set.

Step-by-Step Solution

Answer

The inequalities are: \( x + y \leq 100 \), \( y \geq 20 \), and \( x \geq y \).

Step 1: Define the Variables

Let \( x \) represent the number of fiction books and \( y \) represent the number of nonfiction books.

Step 2: Set Up Inequalities for Total Books

The company publishes a total of no more than 100 books. Therefore, the inequality is \( x + y \leq 100 \).

Step 3: Set Up Inequality for Nonfiction Books

There are at least 20 nonfiction books published. Therefore, the inequality is \( y \geq 20 \).

Step 4: Set Up Inequality for Fiction Books Compared to Nonfiction

The company publishes at least as many fiction books as nonfiction books. Therefore, the inequality is \( x \geq y \).

Step 5: Combine the Inequalities into a System

The system of inequalities that describes the number of books is: 1. \( x + y \leq 100 \)2. \( y \geq 20 \)3. \( x \geq y \).

Step 6: Graph the Solution Set

To graph the solution set, plot each inequality on the coordinate plane.1. For \( x + y = 100 \), draw a line from (100,0) to (0,100).2. For \( y = 20 \), draw a horizontal line at \( y = 20 \).3. For \( x = y \), draw a line through the origin with a slope of 1.Shade the region that satisfies all inequalities: below or on the line \( x+y=100 \), above the line \( y=20 \), and to the right of or on the line \( x=y \).

Key Concepts

Fiction and Nonfiction Books

Books come in two main categories: fiction and nonfiction. Fiction books spring from the imagination and include genres like fantasy, mystery, and romance. These stories are invented by the author and explore characters or events that aren't real. Nonfiction books, on the other hand, are based on facts and real events. They cover a wide range of topics, including biography, history, and science.

In the context of publishing, knowing how many books of each type are produced can be crucial. Publishers often have to strike a balance between the two to cater to different audiences. This balance can depend on market demands and the publisher's goals curated for specific types of readers.

In exercises involving systems of inequalities, focusing on fiction and nonfiction books provides a practical way to understand these mathematical concepts. By considering how many books of each type need to be published, students can apply theoretical math to real-world publishing scenarios.

Graphing Inequalities

Graphing inequalities involves visually representing the solutions to inequality equations on a coordinate plane. When dealing with systems of inequalities, like in this book publishing scenario, multiple inequalities are represented and the solution set needs to satisfy all simultaneously.

Let's explore how to approach graphing the inequalities from the exercise:

• The inequality \(x + y \leq 100\) creates a boundary line on the graph, dividing the plane into two regions. You draw the line connecting the intercept points (100,0) and (0,100) and shade below it to include points that satisfy the inequality.
• For \(y \geq 20\), a horizontal line at \(y = 20\) is drawn across the graph. The region above this line includes the solutions that are valid.
• Lastly, \(x \geq y\) forms a line with a 45-degree slope from the origin, extending through points where fiction and nonfiction books are equal in number. You shade to the right of this line, where fiction books are greater or equal to nonfiction books.

By understanding these graphs and shading techniques, you can effectively visualize the solution set to the system of inequalities.

Linear Inequalities

Linear inequalities are similar to linear equations, but instead of showing equal relationships, they express inequalities \((<, \leq, >, \geq)\) between two expressions. This exercise's inequalities help translate the publishing conditions into mathematical statements that can be manipulated and solved.

Each inequality corresponds to a condition:

• The expression \(x + y \leq 100\) limits the total number of books published. It's an example of a constraint inequality representing a cap on production.
• \(y \geq 20\) dictates a minimum requirement for nonfiction books, reflecting a baseline commitment publishers must meet.
• \(x \geq y\) ensures that fiction books are always published in equal or greater number than nonfiction, showing a preference in production volumes.

These linear inequalities form the foundation of the system. By solving them together, you determine the range of feasible solutions that satisfy all these conditions in the publishing scenario.