Solving Inequalities Determining Maximum Weight For Suitcase Problem

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Introduction

In this article, we will delve into the process of solving a simple inequality problem, which is a fundamental concept in mathematics. Specifically, we will address the problem of determining how much more weight can be added to Li's suitcase without exceeding the 50-pound limit. This problem is represented by the inequality 47.75 + x ≤ 50. Our goal is to find the solution set for x, which represents the additional weight that can be added. Understanding inequalities is crucial in various real-world scenarios, from managing budgets to understanding scientific constraints. This article will not only provide the solution but also explain the underlying principles and steps involved in solving inequalities.

Understanding Inequalities

Before we dive into the specific problem, let's clarify what inequalities are and how they differ from equations. An equation states that two expressions are equal, while an inequality indicates that two expressions are not necessarily equal. Inequalities use symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). The inequality 47.75 + x ≤ 50 signifies that the sum of 47.75 and x is less than or equal to 50. This means that x can take on a range of values, not just a single value as in an equation. To solve this inequality, we need to isolate x on one side of the inequality sign.

Step-by-Step Solution

To solve the inequality 47.75 + x ≤ 50, we need to isolate the variable x. This can be achieved by performing the same operation on both sides of the inequality to maintain the balance, similar to solving equations. Here's a detailed step-by-step solution:

  1. Identify the operation needed to isolate x: In this case, x is being added to 47.75. To isolate x, we need to undo this addition by subtracting 47.75.
  2. Subtract 47.75 from both sides of the inequality: This step ensures that the inequality remains balanced. 47. 75 + x - 47.75 ≤ 50 - 47.75
  3. Simplify both sides: Perform the subtraction on both sides of the inequality. x ≤ 2.25

The solution to the inequality is x ≤ 2.25. This means that the value of x can be any number less than or equal to 2.25. In the context of Li's suitcase, this indicates that Li can add up to 2.25 pounds to her suitcase without exceeding the 50-pound limit.

Interpreting the Solution Set

The solution set for an inequality represents all the values that satisfy the inequality. In this case, the solution set for x ≤ 2.25 includes all real numbers that are less than or equal to 2.25. This can be visualized on a number line, where the solution set is represented by a ray starting at 2.25 and extending to the left, including 2.25 itself. Understanding the solution set is crucial for applying the solution to real-world scenarios. For instance, in Li's case, she needs to ensure that the additional weight she adds to her suitcase does not exceed 2.25 pounds to comply with the weight limit.

Multiple Choice Options

Now, let's examine the multiple-choice options provided and identify the correct one based on our solution:

(A) x ≤ 2.25 (B) x ≤ 2.75 (C) x ≥ 2.25 (D) x ≥ 2.75

Based on our step-by-step solution, the correct answer is (A) x ≤ 2.25. This option accurately represents the solution we derived by isolating x in the inequality.

Common Mistakes to Avoid

When solving inequalities, it's essential to avoid common mistakes that can lead to incorrect solutions. Here are some common pitfalls and how to avoid them:

  • Forgetting to reverse the inequality sign when multiplying or dividing by a negative number: This is a critical rule in inequality operations. If you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. For example, if you have -x < 3, multiplying both sides by -1 gives x > -3.
  • Incorrectly applying the order of operations: Always follow the correct order of operations (PEMDAS/BODMAS) when simplifying expressions in inequalities. This ensures that you perform operations in the correct sequence.
  • Misinterpreting the inequality symbols: Ensure you understand the meaning of each inequality symbol (>, <, ≥, ≤) and how it affects the solution set. For example, x < 3 includes all numbers less than 3, but not 3 itself, while x ≤ 3 includes 3.
  • Not checking the solution: After solving an inequality, it's a good practice to check your solution by substituting a value from your solution set back into the original inequality to ensure it holds true.

By being aware of these common mistakes, you can improve your accuracy in solving inequalities and avoid errors.

Real-World Applications

Inequalities are not just abstract mathematical concepts; they have numerous real-world applications. Understanding inequalities can help in various practical situations. Here are a few examples:

  • Budgeting and Finance: Inequalities can be used to determine how much money can be spent while staying within a budget. For example, if you have a budget of $100 and have already spent $60, you can use the inequality 60 + x ≤ 100 to find the maximum amount (x) you can spend.
  • Weight Limits: As seen in the suitcase problem, inequalities are used to ensure that weight limits are not exceeded, whether for luggage, vehicles, or structures.
  • Speed Limits: Speed limits on roads are expressed as inequalities. For example, a speed limit of 65 mph means that the speed (s) must satisfy the inequality s ≤ 65.
  • Dosage Calculations: In medicine, inequalities are used to determine the safe range for drug dosages. A doctor might prescribe a medication with the instruction to take no more than a certain amount per day, which can be represented as an inequality.
  • Manufacturing: Inequalities are used to set tolerance limits for the dimensions of manufactured parts. For example, a part might need to be within a certain range of sizes to function correctly.

These examples illustrate the practical importance of understanding and solving inequalities in everyday life.

Conclusion

In this article, we have thoroughly examined how to solve the inequality 47.75 + x ≤ 50, which represents the problem of determining the additional weight Li can add to her suitcase without exceeding the 50-pound limit. We have demonstrated a step-by-step solution, identified the correct answer as x ≤ 2.25, and discussed the interpretation of the solution set. Furthermore, we have highlighted common mistakes to avoid when solving inequalities and explored the real-world applications of inequalities. By understanding the principles and techniques involved in solving inequalities, you can apply this knowledge to a wide range of practical problems and improve your mathematical skills.

Mastering inequalities is a valuable skill that extends beyond the classroom, providing a foundation for problem-solving in various aspects of life. Whether you are managing your finances, planning a trip, or working in a technical field, the ability to work with inequalities will prove to be beneficial.