Problem 81 - College Algebra Essentials

Question

Many elevators have a capacity of 2000 pounds. a. If a child averages 50 pounds and an adult 150 pounds, write an inequality that describes when \(x\) children and\(y\) adults will cause the elevator to be overloaded. b. Graph the inequality. Because \(x\) and \(y\) must be positive, limit the graph to quadrant I only. c. Select an ordered pair satisfying the inequality. What are its coordinates and what do they represent in this situation?

Step-by-Step Solution

Answer

The inequality that describes when \(x\) children and \(y\) adults will cause the elevator to be overloaded is \(50x + 150y \leq 2000\). The representative point chosen from the feasible region of the graph is (20, 8), which indicates that the elevator can carry 20 children and 8 adults without being overloaded.

Step 1: Formulate the inequality

To create the inequality, a knowledge of simple arithmetic will be required. The total weight of the children is \(x\) * 50 and the total weight of the adults is \(y\) * 150. For the elevator not to be overloaded, the total weight (children and adults combined) should not exceed 2000 pounds. Therefore, the inequality is \(50x + 150y \leq 2000\).

Step 2: Graph the inequality

To graph the inequality, start by setting values for either \(x\), the number of children, or \(y\), the number of adults, to 0. Then, find the corresponding values of the other variable and plot these points. Connect them with a line, which is the boundary of the region for which the inequality holds. The area below the line (including the line, as the inequality includes equality) in Quadrant I is the feasible region, where the conditions of the inequality are satisfied. Graph it to complete this step.

Step 3: Choose a representative point

Any point in the feasible region or on the boundary line is a solution to the inequality. For example, the point (20,8) which is in the feasible region can be selected. The first number, 20, corresponds to the number of children, \(x\), and the second number, 8, corresponds to the number of adults, \(y\). This means that if there are 20 children and 8 adults, the elevator would still not be overloaded.

Key Concepts

Algebraic Inequalities

Algebraic inequalities are mathematical expressions that involve the symbols <, >, \( \leq \) or \( \geq \) to indicate that one side of the inequality is less than, greater than, less than or equal to, or greater than or equal to the other side.

An inequality like \(50x + 150y \leq 2000\) provides a way to express a range of values for which the inequality holds true. For example, if \(x\) represents the number of children and \(y\) the number of adults, the inequality denotes that the total weight of the group can be equal to or less than 2000 pounds.

To solve an inequality, we look for the value or set of values that satisfy the inequality's conditions. In educational settings, problems often come with contexts, such as the maximum capacity of an elevator, to help visualize and understand the concept. It's essential to interpret the inequality in the context of the real-world scenario it represents.

System of Inequalities

A system of inequalities consists of two or more inequalities that are considered together. Each inequality in the system places its own constraint on the potential solutions, and the solution to the system is the set of all points that satisfy all the inequalities simultaneously.

When graphing a system of inequalities, we find the feasible region that satisfies all constraints. This area represents all the possible solutions to the system. For the elevator scenario, imagine there are multiple constraints like weight limits and the number of individuals. Each condition forms its own inequality, and the overlapped area on the graph where all these conditions are met would illustrate the range of possibilities where all individuals can safely ride the elevator without surpassing its capacity.

Graphical Representation of Inequalities

Graphically representing inequalities helps to visualize the solutions and understand the relationship between variables. The process typically involves plotting the line that represents the boundary of the inequality. This line divides the coordinate plane into two halves.

In our example, the line \(50x + 150y = 2000\) will be the boundary. The inequality \(50x + 150y \leq 2000\) tells us that the solutions are not only the points on the line but also all the points that lie on one side of the line—in this case, below it, as we are dealing with a 'less-than-or-equal-to' inequality.

To graph the inequality, start by finding points that lie on the line, then shade the region that satisfies the inequality. Remember to include the line itself if the inequality symbol includes equality (as indicated by \(\leq\) or \(\geq\)). The resulting shaded area represents all possible combinations of \(x\) and \(y\) that won’t overload the elevator. This method provides a clear and immediate visual interpretation of the feasible solutions.