Problem 71 - Precalculus: Mathematics for Calculus

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: \( f + n \leq 100 \), \( n \geq 20 \), \( f \geq n \).

Step 1: Define Variables

Let \( f \) represent the number of fiction books, and \( n \) represent the number of nonfiction books the company publishes in a year.

Step 2: Total Books Constraint

The total number of books published, both fiction and nonfiction, must not exceed 100. This can be expressed as: \[ f + n \leq 100 \]

Step 3: Nonfiction Minimum Constraint

The company publishes at least 20 nonfiction books annually. This constraint is represented as: \[ n \geq 20 \]

Step 4: Fiction vs. Nonfiction Constraint

The number of fiction books must be at least as much as the number of nonfiction books. This results in the inequality: \[ f \geq n \]

Step 5: Graph the System of Inequalities

To graph these inequalities, plot each on the coordinate plane with \( f \) on the y-axis and \( n \) on the x-axis:- The line for \( f + n = 100 \) (solid line, since the inequality includes 100) and shade below it.- A vertical line at \( n = 20 \) to indicate the region where \( n \geq 20 \), and shade to the right.- The line for \( f = n \) (dashed line altering the shading to keep \( f \geq n \)), and shade above.The intersection of these shaded areas represents the solution set.

Key Concepts

Graphing Inequalities

Understanding how to graph inequalities can help you visualize solutions for complex problems. In our exercise involving the publishing company, we need to graph three inequalities. These inequalities represent limits or boundaries on the possible number of fiction and nonfiction books that can be published in a year. The boundary lines found in these inequalities are crucial.

• The line for the total number of books, \( f + n = 100 \), is drawn as a solid line because the actual value of 100 is included in the possible solutions.
• Next, we have a vertical line at \( n = 20 \), as a solid line too, to show that the company publishes at least 20 nonfiction books. So, any solution must be to the right of this line.
• The equation \( f = n \) is graphed as a dashed line because it sets a condition rather than a limit. The company publishes at least as much fiction as nonfiction, so solutions lie above this line.

The intersection of these shaded regions forms the solution set, where the company can publish books within its policy constraints.

Linear Inequalities

Linear inequalities are like regular linear equations, but they show relationships of more than, less than, and sometimes equal values between two expressions. In our exercise, each inequality reflects a different policy enforced by the publishing company.

• For instance, \( f + n \leq 100 \) shows that the sum of fiction \( f \) and nonfiction \( n \) books must be no more than 100. This inequality deals with the overall limit on book production.
• The constraint \( n \geq 20 \) is straightforward—at least 20 nonfiction books must be published. So, \( n \) cannot be less than 20.
• The last inequality \( f \geq n \) clarifies that the number of fiction books is equal to or exceeds that of nonfiction books. Together, these inequalities form a system of linear inequalities, representing real-world production limits. Learning to interpret and solve these can be quite useful, especially in management and production scenarios.

Constraint Equations

Constraint equations are a powerful tool for any system where resources are limited or specific regulations must be adhered to. In the case of our publishing company, constraints help frame the operational possibilities:

• The equation \( f + n \leq 100 \) acts as a constraint, setting an upper limit on the total number of books produced.
• The inequalities \( n \geq 20 \) and \( f \geq n \) further refine what combinations of fiction and nonfiction books are possible, offering clarity about what the company can and cannot do.
• To effectively harness these constraints, graphs are used. By visualizing these constraints, decision-makers can clearly see the feasible region. This gives them a foundational understanding, helping make informed decisions on resource allocation and strategy adjustments.

With constraint equations, planning becomes methodical, offering a structured representation of conditions that must be satisfied.