Think about the differences between equations and inequalities. For what purpose do you use each one? How is solving for them the same and how is it different?

Question

Create and share a real-world scenario for which you will create an inequality and equation that requires at least two steps to solve. After you have created and shared your own scenario, solve the scenarios shared by two of your peers.

In your response to each post:

Write the solution to both the inequality and the equation.

Explain how the inequality and the equation work together in the scenario.

Answer 1

Differences Between Equations and Inequalities:

Equations state that two expressions are equal. They are used when we are looking for a specific value that satisfies the condition. For example, if we think of equations, they can represent scenarios where we are determining exact quantities, like price or distance.

Inequalities, on the other hand, express a relationship where one side is not necessarily equal to the other; instead, it can be greater than, less than, greater than or equal to, or less than or equal to another expression. Inequalities are useful in situations where there are limits or ranges, such as budgeting, minimum requirements, or maximum capacities.

Similarities in Solving:

• Both equations and inequalities involve manipulating the same types of mathematical expressions (addition, subtraction, multiplication, division).
• Both can be solved using similar algebraic techniques.

Differences in Solving:

• When solving inequalities, if you multiply or divide both sides by a negative number, you must flip the inequality sign.
• Equations yield a specific solution, whereas inequalities often yield a range of solutions.

Real-World Scenario:

Imagine you're planning a charity event and you need to set up the seating. You have a maximum of 120 seats available, and each ticket sells for $25.

• Inequality: Let \( x \) represent the number of tickets sold. The total revenue must be at least $500: \[ 25x \geq 500 \]

• Equation: If you sell all the tickets, you'll have: \[ 25x = 3000 \] for an event where 120 tickets sold would yield $3000 in total revenue.

Solving the Scenarios:

1. Inequality:

   \[ 25x \geq 500 \]

   Divide both sides by 25:

   \[ x \geq 20 \]

   This means you need to sell at least 20 tickets to meet your funding goal.

2. Equation:

   \[ 25x = 3000 \]

   Divide both sides by 25:

   \[ x = 120 \]

   This means you need to sell all 120 tickets to achieve a total revenue of $3000.

Explanation of How They Work Together:

In this scenario, the inequality tells you the minimum number of tickets you need to sell to ensure that you raise at least $500 for the charity, while the equation gives you the total revenue you would achieve by selling all available tickets (120 in this case). Together, they illustrate a feasible plan for ticket sales and help in budgeting for the event. The inequality ensures the event does not fall below a certain financial threshold, while the equation provides a maximum potential earnings scenario.