1-Variable Equations & Inequalities Discussion

Question

1-Variable Equations & Inequalities Discussion

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?

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.
Select the link to access the 1-Variable Equations & Inequalities Discussion.

Answer 1

Bot GPT 4-o mini answered
1 hour ago

Understanding Equations vs. Inequalities

Equations represent a relationship in which two expressions are equal, while inequalities show how one expression is greater than, less than, or not equal to another. We typically use equations when we want to find a specific value or solution, while inequalities help us understand ranges of possible solutions or limits.

Similarities in Solving:

Both equations and inequalities can often be solved using similar algebraic techniques such as addition, subtraction, multiplication, and division.
The goal in both cases is to isolate the variable.

Differences in Solving:

When solving inequalities, if you multiply or divide both sides by a negative number, you must reverse the inequality sign.
Equations yield a specific solution (like x = 3), while inequalities provide a range of solutions (like x > 3).

Real-World Scenario

Scenario: A gym membership costs $30 per month, and there's a one-time registration fee of $50. A person wants to budget for this gym membership but has a limit of $200 to spend in the first month.

Equation: Let $ x $ be the number of months the person plans to stay a member. \[ 30x + 50 = 200 \]
Inequality: \[ 30x + 50 \leq 200 \]

Solving the Scenarios

Solving the Equation: \[ 30x + 50 = 200 \] Subtract 50 from both sides: \[ 30x = 150 \] Divide by 30: \[ x = 5 \]
- The person can afford to be a member for exactly 5 months at $200.
Solving the Inequality: \[ 30x + 50 \leq 200 \] Subtract 50 from both sides: \[ 30x \leq 150 \] Divide by 30: \[ x \leq 5 \]
- The person can afford to be a member for up to 5 months without exceeding their budget.

Explanation of How They Work Together

In this scenario, the equation provides a specific point where the budget equals the total cost, while the inequality indicates the maximum time the person can sustainably afford the membership without exceeding their budget. Both the equation and the inequality reflect the relationship between the monthly membership cost, the initial fee, and the person's budget, helping to guide financial planning for the gym membership.