Problem 61 - Introductory Algebra for College Students

Question

Promoters of a rock concert must sell at least \(25,000\) tickets priced at \(\$ 35\) and \(\$ 50\) per ticket. Furthermore, the promoters must take in at least \(\$ 1,025,000\) in ticket sales. Write and graph a system of inequalities that describes all possibilities for selling the \(\$ 35\) tickets and the \(\$ 50\) tickets.

Step-by-Step Solution

Answer

The system of inequalities representing the possible combinations for selling the $35 and $50 tickets is given by \(x + y \geq 25000\) and \(35x + 50y \geq 1025000\), or in terms of y: \(y \geq 25000 - x\) and \(y \geq 20500 - 0.7x\).

Step 1: Construct the inequalities

The first condition states that at least 25,000 tickets must be sold, this can be written as: \(x + y \geq 25000\). The second condition, which requires a total earning of at least $1,025,000, can be expressed as \(35x + 50y \geq 1025000\).

Step 2: Solve for \(y\)

In order to graph the inequalities, solve both equations for \(y\). The first equation becomes \(y \geq 25000 - x\) and the second equation becomes \(y \geq 20500 - 0.7x\).

Step 3: Graph the inequalities

To graph the inequalities, sketch the lines \(y = 25000 - x\) and \(y = 20500 - 0.7x\). The area above each line represents the solutions for each inequality. The overlapping area above both lines represents the set of solutions to both inequalities. Note that since we are dealing with a real-world context, only positive values of \(x\) and \(y\) (representing amounts of tickets) are relevant.

Key Concepts

Graphing Inequalities

Understanding how to graph inequalities is essential when dealing with systems that involve multiple constraints. With the example of the rock concert promoters, we see two key inequalities that need to be represented graphically.

• The first inequality, representing the minimum number of tickets needed, can be written as \(y \text{-}geq 25000 - x\). On a graph, this would translate to a line where the area above represents potential solutions.
• The second inequality corresponds to the minimum revenue required, expressed as \(y \text{-}geq 20500 - 0.7x\). This creates another line on the graph, where again, the solution set is the area above the line.

When graphing, it's crucial to plot both lines accurately on a graph with both axes representing the number of tickets at the given prices. The intersecting area above both lines shows the feasible options for the number of \(35 and \)50 tickets that can be sold to meet both conditions. Remember, practical solutions only fall in the first quadrant since the number of tickets sold cannot be negative.

Algebraic Inequalities

Algebraic inequalities play a vital role in modeling real-world situations where conditions are not fixed but have to satisfy particular ranges or limits. The ticket sales scenario involves two inequalities:

• The sum of \(35 tickets \(x\) and \)50 tickets \(y\) must be at least 25,000: \(x + y \text{-}geq 25000\).
• The total revenue from both tickets must be no less than $1,025,000: \(35x + 50y \text{-}geq 1025000\).

In solving these algebraic inequalities, we manipulate the conditions algebraically to express one variable in terms of another, making them easier to graph. This often involves subtracting one term from both sides of the inequality or dividing by a coefficient, always remembering if we multiply or divide by a negative number, the inequality sign reverses. In this case, the solutions involve finding any combination of \(x\) and \(y\) that meets both sales and revenue conditions, highlighting the significance of understanding how to manipulate and interpret inequalities in algebra.

Linear Programming

Linear programming is a mathematical method used for optimizing a certain objective, subject to given constraints, and often involves systems of inequalities. In our concert tickets example, the promoters could use linear programming to maximize their profit while adhering to the constraints of ticket sales and revenue.

• The objective function, though not explicitly given, may be to maximize total revenue: \(R = 35x + 50y\).
• The constraints are represented by the two inequalities from the exercise.

Using methods of linear programming, we identify the feasible region on the graph, which is both the solution set to all constraints and the area of interest for optimization. The optimal solution is usually located at the vertices, or the corners, of the feasible region. In this scenario, promoters would look for the point within the feasible region that maximizes their revenue function, ensuring it satisfies both the number of tickets and total revenue constraints.