Problem 44 - Algebra 1

Question

Write an equation or an inequality to model the real-life situation. The product of \(\$ 25\) and the number \(m\) of club memberships is greater than or equal to \(\$ 500.\)

Step-by-Step Solution

Answer

The inequality that models this real-life situation is \(m \geq 20\)

Step 1: Identify the scenario

Read through the problem and identify the key details. The problem provides the cost of each club membership and the total amount raised.

Step 2: Formulate the inequality

The total amount raised (\$500) must be less than or equal to the product of the cost per club membership (\$25) and the number of memberships sold (\(m\)). This can be written as an inequality: \(25m \geq 500\)

Step 3: Simplify the inequality

The inequality can be simplified by diving both sides by 25. The simplified inequality is: \(m \geq 20\)

Key Concepts

Writing Inequalities

Understanding how to write inequalities is crucial when translating real-life situations into mathematical language. It involves recognizing the relationship between quantities and representing them correctly. In our exercise, the situation describes a relationship where the total amount of money raised through club memberships is at least \(500. Here, each membership costs \)25, and we need to find the minimum number of memberships, represented by m, to meet or exceed this amount.

To write this as an inequality, we identify that the product of \(25 and m should be no less than \)500, leading to the inequality: \(25m \geq 500\). Notice how the inequality symbol '\(\geq\)' denotes 'greater than or equal to', capturing the requirement for the amount raised to be at least $500. It's important to choose the right inequality symbol to represent the scenario accurately.

Algebraic Expressions

Algebraic expressions are a way to represent real-world quantities and their relationships using variables and numbers. In this case, we have the expression \(25m\), where the variable m stands for the unknown quantity—the number of club memberships. The number 25 represents the fixed cost per membership.

An important aspect of algebraic expressions is understanding how to use variables to stand in for unknowns. Once we have an expression like \(25m\), we can perform various operations on it, such as simplifying or combining it with other expressions, while maintaining the relationships initially set out in the problem.

Solving Inequalities

Solving inequalities is about finding the set of all possible values that satisfy the inequality condition. For our example, the inequality \(25m \geq 500\) is solved by isolating m on one side. This is achieved by dividing both sides of the inequality by 25, the coefficient of m, which simplifies the inequality to \(m \geq 20\).

This result tells us that at least 20 memberships need to be sold. It's essential when solving inequalities to remember that if we multiply or divide both sides of an inequality by a negative number, we must reverse the inequality sign. However, in this instance, we divided by a positive number, so the direction of the inequality remained the same.