alina spent no more than $45 on gas

2 min read 30-11-2024

Meta Description: Alina had a gas budget! Learn how to solve word problems involving inequalities to figure out the maximum amount of gas Alina could buy. This step-by-step guide breaks down the problem and teaches you how to set up and solve inequalities. Perfect for math students of all levels!

Understanding Alina's Spending Limit

Alina's situation presents a classic math word problem involving inequalities. The core information is this: Alina spent no more than $45 on gas. This means her spending was less than or equal to $45. We can use this information to create a mathematical inequality and solve for potential amounts of gas she could have purchased.

Setting up the Inequality

Let's use 'g' to represent the amount Alina spent on gas (in dollars). The phrase "no more than $45" translates directly into the inequality:

g ≤ 45

This inequality states that the amount Alina spent on gas (g) is less than or equal to 45.

Visualizing the Solution

We can visualize this inequality on a number line. The solution includes all values of 'g' from 0 up to and including 45. This represents all the possible amounts Alina could have spent on gas, within her budget.

[Insert image here: A number line showing a shaded region from 0 to 45, with a closed circle at 45 to indicate inclusion.] Alt text: Number line showing the solution to the inequality g ≤ 45.

Exploring Different Scenarios

Let's explore some possible scenarios based on different gas prices:

Scenario 1: Gas costs $3 per gallon.

If gas costs $3 per gallon, and Alina spent exactly $45, we can find how many gallons she bought:

$45 ÷ $3/gallon = 15 gallons

In this case, Alina bought 15 gallons of gas. She stayed within her budget.

Scenario 2: Gas costs $2.50 per gallon.

If gas costs $2.50 per gallon, and Alina spent $40, we can calculate the gallons purchased:

$40 ÷ $2.50/gallon = 16 gallons

Here, Alina bought 16 gallons and still stayed under her $45 budget.

Scenario 3: Gas costs $4 per gallon.

If gas cost $4 per gallon, and Alina spent her maximum of $45, she could buy:

$45 ÷ $4/gallon = 11.25 gallons

In this scenario, she might have purchased 11.25 gallons (assuming gas stations allow for partial gallons).

Solving for Unknown Variables

These scenarios illustrate how the inequality helps us understand the range of possibilities. We can easily adjust the gas price to explore different amounts Alina could have purchased while remaining within her budget constraint.

Practical Applications and Further Exploration

This simple word problem demonstrates a fundamental concept in algebra: using inequalities to represent real-world constraints. Understanding inequalities is crucial for solving many practical problems, from budgeting to calculating material needs for projects.

For further exploration, consider these questions:

What if Alina had a different budget? How would changing the maximum spending amount affect the inequality and the possible solutions?
What if the gas price fluctuated? How could you incorporate changing gas prices into the problem to create a more dynamic model?
Can you create a similar word problem involving another type of purchase and a spending limit?

By tackling these questions, you can deepen your understanding of inequalities and their applications in everyday situations. Understanding how to solve this seemingly simple problem provides a strong foundation for more complex mathematical scenarios.