Problem 40 - Prealgebra

Question

There are more than \(32,150\) high school girls basketball and track programs in the United States. If there are \(15,089\) girls track programs, write and solve an inequality to determine the number of girls basketball programs.

Step-by-Step Solution

Answer

There are more than 17,061 girls basketball programs.

Step 1: Define the Variables

Let the number of girls basketball programs be represented by the variable \( b \). Our goal is to determine the inequality that can be used to find \( b \).

Step 2: Set Up the Inequality

There are more than 32,150 high school girls basketball and track programs combined. This can be written as the inequality: \( b + 15,089 > 32,150 \).

Step 3: Solve the Inequality

Subtract 15,089 from both sides of the inequality to isolate \( b \) on one side. This gives us: \( b > 32,150 - 15,089 \).

Step 4: Calculate the Solution

Perform the calculation: \( 32,150 - 15,089 = 17,061 \). Thus, the inequality becomes \( b > 17,061 \).

Key Concepts

Inequality Solving

Inequality solving in mathematics involves determining the set of possible solutions that satisfy the conditions of an inequality. Unlike equations, inequalities do not represent exact equal amounts, but rather a range of possible values. In this exercise, we set up an inequality to find out whether there are more girls basketball programs than a specific number.

To solve an inequality like the one provided, start by simplifying both sides as much as possible. You can manage this by using mathematical operations to isolate the variable of interest on one side. In this case, the inequality starts as:

• \( b + 15,089 > 32,150 \)

After performing the subtraction operation, the simplified inequality \( b > 17,061 \) shows the number of basketball programs must be greater than 17,061 for the combined total to exceed 32,150.

Make sure to always do the same operation to both sides of the inequality to keep it balanced.

Variable Definition

Variable definition is a crucial first step in solving any mathematical problem, especially when dealing with inequalities. In this exercise, the variable "\( b \)" was used to represent the unknown quantity—in this case, the number of girls basketball programs.

Defining a variable helps transform a word problem into a mathematical statement which can be solved using mathematical operations. It acts as a placeholder for the unknown value we are trying to find.

When choosing a variable:

• Pick a letter that can remind you of what you're solving for; "\( b \)" stands for basketball programs.
• Clearly state what the variable represents at the beginning of the problem-solving process.

A clear definition of your variables helps in setting up the correct equation or inequality needed to solve the problem.

Mathematical Operations

Mathematical operations are the tools we use to manipulate equations and inequalities to isolate variables and find solutions. In this problem, subtraction was used to solve the inequality:

• We began with \( b + 15,089 > 32,150 \).
• To isolate \( b \), subtract 15,089 from both sides, resulting in \( b > 32,150 - 15,089 \).

This highlights the importance of applying the correct mathematical operations to both sides of an inequality to maintain its integrity.

Once the subtraction is performed, continuing with the equality yields the solution. The arithmetic operation gave us the value \( 17,061 \), altering the simplified inequality to \( b > 17,061 \).

Always remember that with inequalities, when multiplying or dividing by a negative number, you need to flip the inequality sign.

Problem-Solving Strategies

Problem-solving strategies in mathematics help in efficiently arriving at a solution. When tackling any math problem, particularly those involving inequalities, it's essential to break the problem into manageable steps. Here are some strategies used in the given exercise:

• **Understand the Problem:** First, interpret the problem to understand what is asked. Recognize that you need to find how many basketball programs there can be such that their total with track programs exceeds 32,150.
• **Develop a Plan:** Define variables carefully and set up an inequality that represents the problem statement.
• **Execute the Plan:** Perform necessary mathematical operations (like subtraction) consistently on both sides of the inequality to isolate the variable.
• **Review and Interpret the Solution:** Once calculations are done, interpret the mathematical statement \( b > 17,061 \) as the solution to understand its real-world implication.

The above strategies ensure a structured approach to solving math problems, thereby reducing errors and making solutions easier to verify.