Gas Cost Inequality: Find Gallons Purchased At Two Stations

Have you ever wondered how to calculate your total gas expenses when purchasing fuel at different prices? This article will guide you through formulating an inequality to represent such scenarios. We'll use a real-world example of Alina's road trip expenses to illustrate the concept. Understanding inequalities is crucial not only in mathematics but also in everyday financial planning. So, let's dive in and explore how to set up an inequality that models Alina's gas expenses.

Understanding the Problem: Alina's Road Trip

To begin, let's understand the problem. Alina spent no more than $45 on gas for her road trip. This "no more than" is a crucial phrase indicating that we're dealing with an inequality rather than a simple equation. She purchased gas at two different stations: the first charged $3.50 per gallon, and the second charged $4.00 per gallon. Our goal is to create an inequality that relates the number of gallons she bought at the first station to her total expenditure. This involves translating the word problem into a mathematical expression, a skill vital for solving many real-world problems. We need to define variables, understand the constraints, and formulate an inequality that captures the given information. By breaking down the problem into smaller parts, we can methodically construct the inequality.

Defining Variables: The Key to Formulation

Before we jump into forming the inequality, we need to define our variables. This is a critical step in translating any word problem into a mathematical equation or inequality. Let's use 'x' to represent the number of gallons of gas Alina bought at the first station, which charged $3.50 per gallon. The problem does not specify the amount of gas she purchased at the second station, so we will focus on the relationship involving 'x'. Defining variables clearly helps us to organize the information and construct the inequality accurately. It allows us to represent unknown quantities with symbols and manipulate them algebraically. Without clearly defined variables, the process of formulating an inequality can become confusing and prone to errors. So, taking the time to define variables is a fundamental step in problem-solving.

Constructing the Inequality: Putting the Pieces Together

Now that we have defined our variable, let's start constructing the inequality. We know that Alina spent $3.50 per gallon at the first station, so the total cost for the gas from the first station is 3.50x. Since we don't have information about the second station's gas purchase in relation to 'x,' we'll focus on what we know: Alina spent no more than $45 in total. This "no more than" translates mathematically to "less than or equal to" (≤). Thus, we can express the inequality as:

3. 50x ≤ 45

This inequality represents the constraint on Alina's spending. The total cost of gas from the first station (3.50x) must be less than or equal to $45. This simple inequality captures the essence of the problem and allows us to analyze the possible values of 'x'. By setting up the inequality correctly, we can now solve it to determine the maximum number of gallons Alina could have purchased at the first station. The ability to translate real-world scenarios into mathematical expressions is a valuable skill in various fields, from finance to engineering.

Solving the Inequality: Finding the Limit

To determine the maximum number of gallons Alina could have purchased at the first station, we need to solve the inequality 3.50x ≤ 45. We do this by isolating 'x' on one side of the inequality. To isolate 'x', we divide both sides of the inequality by 3.50:

x ≤ 45 / 3.50

Performing the division, we get:

x ≤ 12.857

Since Alina cannot buy a fraction of a gallon in practical terms (unless the gas station measures partial gallons, which is unlikely), we round down to the nearest whole number. Therefore, Alina could have bought a maximum of 12 gallons of gas at the first station. Solving the inequality provides a concrete answer to the problem, giving us a limit on the number of gallons Alina could have purchased. This demonstrates the practical application of inequalities in real-world scenarios. Understanding how to solve inequalities is essential for making informed decisions in various situations involving constraints and limitations.

Interpreting the Solution: Real-World Context

Our solution, x ≤ 12, tells us that Alina could have purchased a maximum of 12 gallons of gas at the first station while staying within her $45 budget. This is a crucial interpretation in the context of the problem. It's important to remember that in real-world situations, the mathematical solution needs to make practical sense. Alina couldn't have bought a negative amount of gas, nor could she practically buy a fractional amount beyond what's measurable at a gas pump. The inequality provides a range of possible values for 'x', but the real-world context helps us narrow down the solution to the most reasonable answer. Interpreting the solution in context is a vital step in problem-solving, as it ensures that our mathematical results are meaningful and applicable to the situation at hand. This skill is essential in various fields, from science to economics, where mathematical models are used to make predictions and decisions.

Expanding the Problem: What If We Knew More?

Let's consider how the inequality would change if we had additional information. For example, what if we knew the number of gallons Alina purchased at the second station? Let's say she bought 'y' gallons at the second station, which charged $4.00 per gallon. The cost for gas at the second station would be 4.00y. Now, our inequality would need to incorporate this new variable:

3. 50x + 4.00y ≤ 45

This new inequality represents the total cost of gas from both stations. If we knew the value of 'y', we could solve for 'x' to find the maximum number of gallons Alina could have purchased at the first station, given her purchase at the second station. This illustrates how inequalities can be adapted and expanded to accommodate more complex scenarios. Adding more variables and constraints allows us to create more realistic models of real-world situations. This ability to expand and adapt mathematical models is a powerful tool in problem-solving and decision-making.

Why Inequalities Matter: Real-World Applications

Inequalities are more than just mathematical concepts; they are powerful tools for modeling real-world constraints and making informed decisions. In our example, the inequality helped us determine the maximum amount of gas Alina could purchase within her budget. This type of problem-solving is applicable in various scenarios, from budgeting and finance to engineering and logistics. Inequalities are used to optimize resources, set limits, and analyze constraints in numerous industries. For instance, a company might use inequalities to determine the maximum production output given limited resources, or a civil engineer might use them to calculate the maximum load a bridge can support. Understanding inequalities empowers us to make informed decisions and solve complex problems in everyday life and professional settings. This highlights the importance of mathematical literacy in navigating the modern world.

Conclusion: Mastering Inequalities

In conclusion, formulating an inequality to represent real-world scenarios, like Alina's gas expenses, involves defining variables, translating word problems into mathematical expressions, and interpreting the solution in context. We've seen how the inequality 3.50x ≤ 45 effectively models Alina's spending limit at the first gas station. By solving this inequality, we determined that Alina could purchase a maximum of 12 gallons of gas while staying within her budget. This process highlights the practical application of inequalities in everyday problem-solving. Mastering inequalities is not just about mathematical proficiency; it's about developing critical thinking skills that are essential for making informed decisions in various aspects of life. We encourage you to explore further resources and practice applying inequalities to different scenarios to enhance your understanding and problem-solving abilities. For further learning, you can explore resources like Khan Academy which offers comprehensive lessons and practice exercises on inequalities.