Problem 51 - Precalculus Mathematics for Calculus

Question

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 system of inequalities is \( x + y \leq 100 \), \( x \geq 20 \), and \( y \geq x \).

Step 1: Identify Variables

Let \( x \) represent the number of nonfiction books, and let \( y \) represent the number of fiction books published by the company each year.

Step 2: Write Inequality for Total Number of Books

The company publishes no more than 100 books per year, so we can represent this with the inequality: \( x + y \leq 100 \).

Step 3: Write Inequality for Minimum Nonfiction Books

According to the conditions, at least 20 of the books are nonfiction, which is expressed as: \( x \geq 20 \).

Step 4: Write Inequality for Fiction Books

The company always publishes at least as much fiction as nonfiction. This is represented by: \( y \geq x \).

Step 5: Compile the System of Inequalities

Putting all the conditions together, the system of inequalities that describes the situation is:1. \( x + y \leq 100 \)2. \( x \geq 20 \)3. \( y \geq x \).

Step 6: Graph the Solution Set

To graph the system, plot the lines corresponding to the equations \( x + y = 100 \), \( x = 20 \), and \( y = x \) on a coordinate grid.- Shade the region below the line \( x + y = 100 \) (including the line) to satisfy \( x + y \leq 100 \).- Shade the region to the right of the line \( x = 20 \) to satisfy \( x \geq 20 \).- Shade the region above the line \( y = x \) to satisfy \( y \geq x \).- The feasible region is where all these shaded areas overlap, representing the possible combinations of nonfiction and fiction books.

Key Concepts

Graphing Inequalities

Graphing inequalities involves representing a range of solutions on a coordinate grid. For a system of inequalities, each linear inequality divides the plane into a half-plane. The solution to the inequality is the set of points in this half-plane. To graph an inequality:

• First, graph the corresponding equality as a line. For example, for the inequality \( x + y \leq 100 \), you first graph the line \( x + y = 100 \).
• Use a solid line if the inequality contains \( \leq \) or \( \geq \). Use a dashed line for \( < \) or \( > \).
• Then shade the area where the inequality holds true. For instance, below or above the line depending on the inequality sign.

In the exercise, graphing the inequalities \( x + y \leq 100 \), \( x \geq 20 \), and \( y \geq x \) results in shaded regions where these conditions are met. The final graph showcases where all conditions overlap, forming the feasible solution area.

Linear Inequalities

Linear inequalities involve expressions with two variables where the variables satisfy a certain range of values. They can be presented in the form of equations but with inequality signs such as \( \leq \), \( \geq \), \( < \), or \( > \) instead.
This means instead of only finding a single line, like in linear equations, you find a whole region.
In the given problem:

• \( x + y \leq 100 \) represents that the total books should be no more than 100.
• \( x \geq 20 \) ensures the nonfiction books are not less than 20.
• \( y \geq x \) establishes that there are at least as many fiction books as nonfiction.

This system of inequalities describes various potential outcomes for how many fiction and nonfiction books can be published yearly by the company, as long as they meet all given conditions.

Feasible Region

The feasible region is the area where all inequalities in a system overlap on the graph. It represents all the potential solutions that satisfy every inequality at the same time.

• To identify the feasible region, first graph each inequality carefully.
• Shade the correct side of each line according to the inequality direction.
• The overlapping shaded area from all inequalities is the feasible region.

In this exercise, the feasible region shows different combinations of nonfiction and fiction books the company can publish yearly. This region is the solution to the system of inequalities, clarifying which mixes of book genres respect all the imposed policies.