Problem 37 - Algebra 1: Concepts and Skills

Question

Use the following information. You can work a total of no more than 20 hours per week at your two jobs. Baby-sitting pays 5 dollars per hour, and your job as a cashier pays 6 dollars per hour. You need to earn at least 90 dollars per week to cover your expenses. Write a system of inequalities that shows the various numbers of hours you can work at each job.

Step-by-Step Solution

Answer

The system of inequalities representing the feasible number of hours that can be worked at each job is: \(\begin{cases}x + y \leq 20 \ 5x + 6y \geq 90\end{cases}\), where \(x\) is the number of hours worked as a baby-sitter and \(y\) is the number of hours worked as a cashier.

Step 1: Define the Variables

Let \(x\) represent the number of hours worked as a baby-sitter and \(y\) represent the number of hours worked as a cashier.

Step 2: First constraint

The total number of hours worked at both jobs cannot exceed 20 hours. This can be expressed as: \(x + y \leq 20\)

Step 3: Second constraint

The total amount earned must be at least 90 dollars. Given the babysitting job pays 5 dollars per hour and the cashier job pays 6 dollars per hour, this can be expressed as: \(5x + 6y \geq 90\)

Step 4: System of Inequalities

So, the system of inequalities that represent the number of hours that can be worked at each job are: \(\begin{cases}x + y \leq 20 \ 5x + 6y \geq 90\end{cases}\)

Key Concepts

Linear Inequalities

Linear inequalities are mathematical expressions that compare two linear expressions using inequality signs such as "less than or equal to" (\(\leq\)) or "greater than or equal to" (\(\geq\)). These inequalities describe a range of possible solutions instead of one exact solution. In the problem, we faced two inequalities. The first inequality, \(x + y \leq 20\), tells us that the combined hours spent working the two jobs must not exceed 20 hours per week. This means that any solution for \(x\) (hours babysitting) and \(y\) (hours cashiering) should satisfy this maximum hour condition.

The second inequality, \(5x + 6y \geq 90\), represents the required minimum earnings. Here, \(5x\) reflects the earnings from babysitting, and \(6y\) is what you earn at the cashier job. This inequality says that these earnings together should sum up to at least 90 dollars. Together, these inequalities form a system—a dual condition that any potential solution must satisfy.

Variable Definition

Variable definition is crucial when solving systems of inequalities. Defining what \(x\) and \(y\) represent helps in constructing correct mathematical models. In our scenario, we chose \(x\) to denote the number of hours spent babysitting and \(y\) for cashiering. Defining these variables correctly is the first step in turning a real-world problem into a mathematical framework.

Without clear and precise definitions, these variables could easily become confusing, especially when we formulate the inequalities that govern the problem. For example, our definition allows us to understand intuitively why \(x + y \leq 20\) means limiting work hours or why \(5x + 6y \geq 90\) focuses on calculating earnings.

• \(x\) = hours babysitting
• \(y\) = hours cashiering

By maintaining these definitions, we ensure that the problem is framed consistently as we perform calculations and seek solutions.

Constraint Modeling

Constraint modeling involves representing real-world restrictions or limits using mathematical statements. This is often done using linear inequalities. In this exercise, constraints were formed around the total hours available for work and the minimum income requirement.

The first constraint, \(x + y \leq 20\), models the maximum number of hours you can work in a week at both jobs. Each hour beyond 20 is a dead end in terms of finding solutions. This constraint ensures that proposed solutions don't demand more hours than physically possible.

The second constraint, \(5x + 6y \geq 90\), ensures the income requirement is met. This model enforces that any plan involving babysitting and cashier work will feed into sustainable earnings.

When we capture these constraints in inequalities, we quantify and formalize the conditions, making it easier to visualize and solve. Constraint modeling translates practical limitations into a mathematical form that can be analyzed and solved systematically using the tools of algebra.