Problem 19 - Prealgebra

Question

Tomás wants to spend less than \(\$ 100\) for a new soccer ball and shoes. The ball costs \(\$ 24 .\) Write and solve an inequality that gives the amount that Tomás can spend on shoes.

Step-by-Step Solution

Answer

Tomás can spend less than $76 on the shoes.

Step 1: Understanding the Problem

We need to find out how much Tomás can spend on shoes without exceeding his total budget of $100. He already knows the cost of the soccer ball, which is $24.

Step 2: Setup the Inequality

Let's denote the cost of the shoes as \( x \). Tomás wants his total expenditure on the ball and shoes to be less than $100. Thus, the inequality is formulated as follows: \( 24 + x < 100 \).

Step 3: Solving the Inequality

To find \( x \), we need to isolate \( x \) by subtracting 24 from both sides of the inequality: \( x < 100 - 24 \).

Step 4: Calculate the Result

Performing the subtraction gives \( x < 76 \). This means Tomás can spend less than $76 on the shoes.

Key Concepts

Budgeting

Effective budgeting is a crucial skill, especially when managing limited financial resources. It involves planning expenses so that they do not exceed available funds. For example, Tomás has a total budget of $100 to buy a soccer ball and shoes. This means he needs to consider the cost of both items to ensure he stays within this limit.
Tomás's approach includes these steps:

• Identify the total budget available, which is $100.
• Subtract the known cost, the $24 soccer ball, to determine how much is left for the shoes.
• Set a spending limit to ensure the total expenditure remains less than the budget.

By organizing finances like this, Tomás avoids overspending while still meeting his needs. Such planning and careful calculation are what make budgeting an effective financial management strategy.

Solving Inequalities

Solving inequalities is a simple yet powerful mathematical skill. An inequality is similar to an equation, but instead of equality, it shows that two expressions are not equivalent. In our example, Tomás wants his total spending to be less than \(100.
Here is how we solve the inequality:

• Set up an inequality with the known expenses and variables, like we did with \)24 for the ball and \(x\) for the shoes: \(24 + x < 100\).
• Isolate the variable by performing inverse operations, such as subtraction: \(x < 100 - 24\).
• Simplify the expression to find the value range for the variable: \(x < 76\).

Concluding this process, Tomás finds he can spend less than $76 on shoes.
Solving inequalities efficiently requires understanding each step and accurately manipulating mathematical expressions. This is useful not only in personal budgeting but across many applications in everyday life.

Mathematical Reasoning

Mathematical reasoning is about using logic to solve problems effectively. It’s the backbone of working through any mathematical problem, including inequalities. For Tomás's situation, reasoning helps structure the problem and find a solution logically.
Here’s how it applies:

• Recognize the requirement to spend less than a certain amount, which involves comparing total expenses with available funds.
• Decide how to express this comparison using mathematical symbols, forming the inequality \(24 + x < 100\).
• Use logical steps to transform the inequality and deduce the solution for \(x\).

By systematically approaching the initial inequality and processing it through logical steps, **mathematical reasoning** becomes the tool that turns abstract numbers into tangible financial decisions.
Reasoning in this way is a cornerstone of successful problem-solving across all areas of math.