Problem 39 - Algebra 1

Question

Write an equation or an inequality to model the real-life situation. Ben's hourly wage \(b\) at his after school job is \(\$ 1.50\) less than Eileen's hourly wage \(e.\)

Step-by-Step Solution

Answer

\(b = e - 1.50\), where \(b\) represents Ben's hourly wage and \(e\) represents Eileen's hourly wage.

Step 1: Identifying the known information

In the problem, we know that Ben's wage is \(b\) and Eileen's wage is \(e\). Further, we know that Ben's wage is $1.50 less than Eileen's hourly wage, which indicates a subtract operation.

Step 2: Writing the equation

From the information given, we can write an equation that represents the financial relationship between Ben's wage (\(b\)) and Eileen's wage (\(e\)). Ben's wage is $1.50 less than Eileen's which is represented mathematically as \(b = e - 1.50\). This equation models the real-life situation explained in the exercise.

Key Concepts

Real-Life Applications

Understanding how algebraic equations apply in real life is essential for solving everyday problems. In this exercise, we look at a situation involving wages, something most people are familiar with. Here, Ben earns $1.50 less per hour than Eileen. Such scenarios are common in the workplace, where differences in payment can depend on experience or job roles. This context helps us see algebra not just as abstract numbers and symbols, but as tools to describe and analyze real-world situations. Using algebra, we can make informed decisions, estimate future earnings, and better manage our finances.

Inequality Modeling

While inequalities are not directly used in this example, understanding them is crucial. Inequalities help us describe situations where values are not exactly equal, and they often come into play in real-life scenarios. If we were to ask, "Can Ben's wage be close to or greater than Eileen's?", we might write an inequality like \[ b \geq e - 1.50 \]Understanding inequalities allows us to express not only fixed differences but also potential ranges. Exploring such ideas can prepare you for scenarios where terms like 'at least,' 'no more than,' or 'greater than' appear.

Problem-Solving

Algebra is a powerful tool in problem-solving. Here, we broke down the initial problem into understandable parts, identifying known quantities and relationships.

• First, recognize what each variable represents. Ben's wage as \(b\) and Eileen's wage as \(e\).
• Next, understand the relationship, Ben earning $1.50 less than Eileen.
• Finally, express this relationship mathematically as an equation.

This step-by-step method can be applied to various situations, enabling clearer thinking and organized solutions. By systematically tackling each part, we solve problems more effectively.

Mathematical Expressions

Mathematical expressions allow us to clearly communicate mathematical ideas. In this exercise, Ben's wage expressed as \( b = e - 1.50 \) is a simple example of how we use expressions. They involve variables that can change, like \(b\) and \(e\), and constants, such as 1.50, which don't change. When creating expressions:

• Identify variables and their meanings in the context.
• Use mathematical operations (addition, subtraction) to express relationships.
• Ensure the expression accurately reflects the real-world scenario.

Mastery of expressions leads to better understanding and communication of complex ideas.