Problem 83 - College Algebra Essentials

Question

On your next vacation, you will divide lodging between large resorts and small inns. Let \(x\) represent the number of nights spent in large resorts. Let \(y\) represent the number of nights spent in small inns. a. Write a system of inequalities that models the following conditions:You want to stay at least 5 nights. At least one night should be spent at a large resort. Large resorts average \(\$ 200\) per night and small inns average \(\$ 100\) per night. Your budget permits no more than \(\$ 700\) for lodging. b. Graph the solution set of the system of inequalities in part (a). c. Based on your graph in part (b), what is the greatest number of nights you could spend at a large resort and still stay within your budget?

Step-by-Step Solution

Answer

The maximum nights spent at large resorts would depend on the graph drawn, since the third inequality might intersect the first at a non-integer point. Always round down, because nights can't be split. You need to prepare a graph to provide a definitive answer.

Step 1: Translate the Conditions Into a System of Inequalities

The following inequalities can be formed: \n1. The total nights should be at least 5, therefore, \(x + y \geq 5\). \n2. At least one night should be stay at a large resort, which implies \(x \geq 1\). \n3. The sum cost of nights spent at both the places should not exceed $700. Given, large resorts average $200 per night (x nights), and small inns average $100 per night (y nights), denotes as \(200x + 100y \leq 700\).

Step 2: Sketch the Inequalities on a Graph

Sketch a graph with the x-axis representing the number of nights spent at a large resort and the y-axis denoting the number of nights spent at a small inn. The lines \(x + y = 5\), \(x = 1\) and \(200x + 100y = 700\) are then plotted. The solution set will exist within the region satisfies all inequalities.

Step 3: Determine the Maximum Nights at a Large Resort

By looking at the graph, identify the solutions which satisfies all inequalities and note the maximum integer value of x (nights in large resort) within the budget.

Key Concepts

Graphing Inequalities

When working with systems of inequalities, graphing them visually represents the solutions which satisfy all given conditions. Each inequality is represented by a line or boundary on a graph. The shaded area where all such regions overlap indicates all potential solutions. For example:

• The inequality \(x + y \geq 5\) will be represented by the line \(x + y = 5\). All points above or on this line satisfy the inequality.
• The inequality \(x \geq 1\) is depicted by the line \(x = 1\), with solutions lying on the right of this line.
• The inequality \(200x + 100y \leq 700\) forms the line \(200x + 100y = 700\). Solutions include all points on or below this line.

To find the region satisfying all these inequalities simultaneously, we look for the overlapping section of all shaded areas on the graph. This region is the feasible solution area.

Budget Constraints

Budget constraints are important in decision making when resources are limited. They set a boundary within which all activities must fit. If we take the example of choosing between staying in large resorts and small inns:

• Each night at a large resort costs \\(200, while a night at a small inn costs \\)100.
• Your total budget is \$700, creating the constraint \(200x + 100y \leq 700\).

This inequality forms part of the boundary in our system of inequalities. It determines how many nights you can afford at both types of accommodations without exceeding your budget. Staying within budget means selecting a combination of nights for each type of lodging that satisfies this inequality.

Integer Solutions

In many real-world applications, solutions must often be whole numbers. This is especially true when dealing with quantities such as nights or items, which don't divide neatly into fractions. For the given system, you not only need to locate the feasible region based on the graph but also ensure that solutions are integers.After determining the feasible region, look for coordinate points with whole numbers, such as \((x, y)\). These points must satisfy the specific requirements and constraints of the problem. In this case, you need integer solutions because you can't stay a fraction of a night in one location during your vacation.

Algebraic Modeling

Algebraic modeling consists of creating algebraic expressions and equations to represent and analyze real-world situations. In this case, we have modeled a vacation lodging scenario using a system of inequalities:

• \(x + y \geq 5\): Represents the minimum total nights you want to stay.
• \(x \geq 1\): Ensures at least one night at a large resort.
• \(200x + 100y \leq 700\): Represents the maximum budget for lodging.

These inequalities form a model that helps in making decisions about the numbers of nights spent at different types of accommodations within given constraints. By solving this system graphically, we find combinations that meet all requirements, helping to make a realistic and feasible lodging plan for the vacation.