Solving The Inequality 215 - 2 * 1 - 3 ≤ 15 A Step-by-Step Guide

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Introduction

In this article, we will dissect the inequality 215 - 2 * 1 - 3 ≤ 15 to determine its solution set. Inequalities, a fundamental concept in mathematics, are mathematical statements that compare two expressions using symbols such as less than (<<), greater than (>), less than or equal to (≤), and greater than or equal to (≥). Solving inequalities involves finding the range of values that satisfy the given condition. This article provides a step-by-step breakdown of how to solve the inequality, along with a detailed explanation of the concepts involved. Understanding how to solve inequalities is crucial for various mathematical applications and is a core skill tested in numerous standardized exams.

Understanding Inequalities

Before diving into the solution, it's essential to grasp the basics of inequalities. An inequality is a mathematical statement that compares two expressions, indicating that they are not necessarily equal. Unlike equations, which have specific solutions, inequalities typically have a range of solutions. This range represents all the values that make the inequality true. The symbols used in inequalities are:

  • <: Less than
  • >: Greater than
  • ≤: Less than or equal to
  • ≥: Greater than or equal to

Solving inequalities involves similar techniques as solving equations, but with a crucial difference: when multiplying or dividing by a negative number, the direction of the inequality sign must be reversed. This is because multiplying or dividing by a negative number changes the sign of the expressions being compared, thus affecting their relative order.

Step-by-Step Solution

To solve the inequality 215 - 2 * 1 - 3 ≤ 15, we will follow a systematic approach, breaking down the problem into manageable steps.

1. Simplify the Left-Hand Side (LHS)

First, we simplify the left-hand side of the inequality by performing the arithmetic operations. The expression is 215 - 2 * 1 - 3. Following the order of operations (PEMDAS/BODMAS), we perform the multiplication before the subtraction.

2 * 1 = 2

Now, substitute this result back into the expression:

215 - 2 - 3

Next, we perform the subtraction from left to right:

215 - 2 = 213

213 - 3 = 210

So, the simplified left-hand side is 210.

2. Rewrite the Inequality

Now, we rewrite the inequality with the simplified left-hand side:

210 ≤ 15

3. Analyze the Inequality

At this point, we have the inequality 210 ≤ 15. This statement asserts that 210 is less than or equal to 15. However, this is clearly not true. 210 is much greater than 15. Therefore, the inequality 210 ≤ 15 is a false statement.

4. Determine the Solution Set

Since the simplified inequality 210 ≤ 15 is false, there are no values that satisfy the original inequality 215 - 2 * 1 - 3 ≤ 15. This means the solution set is empty. In mathematical notation, the empty set is represented by ∅.

Analyzing the Given Options

Now, let's examine the options provided to see if any of them represent the solution set we found.

A. (-\infty, -2) ∪ (7, \infty)

This option represents the set of all real numbers less than -2 or greater than 7. It is an unbounded interval and does not match our empty solution set.

B. (-\infty, 15) ∪ (7.5, \infty)

This option represents the set of all real numbers less than 15 or greater than 7.5. Like option A, this is an unbounded interval and does not match our empty solution set.

C. [-2, 7]

This option represents the closed interval from -2 to 7, including -2 and 7. This is a bounded interval, meaning it has a defined start and end point. It does not match our empty solution set.

D. [1.5, 7.5]

This option represents the closed interval from 1.5 to 7.5, including 1.5 and 7.5. This is another bounded interval, but it also does not match our empty solution set.

Since none of the provided options represent an empty set, it's crucial to recognize that the correct solution might not always be explicitly listed. In this case, the absence of a matching option reinforces our understanding that the inequality has no solution.

Common Mistakes and How to Avoid Them

When solving inequalities, it's easy to make mistakes if you're not careful. Here are some common pitfalls and how to avoid them:

  1. Forgetting to Reverse the Inequality Sign: One of the most frequent errors is failing to reverse the direction of the inequality sign when multiplying or dividing by a negative number. Always remember this rule to ensure you're finding the correct solution set.
  2. Incorrect Order of Operations: Not following the order of operations (PEMDAS/BODMAS) can lead to incorrect simplification of expressions. Always perform operations in the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction.
  3. Misinterpreting the Solution Set: It's crucial to understand what the solution set represents. For example, an empty set means there are no solutions, while an unbounded interval means the solution extends infinitely in one or both directions.
  4. Algebraic Errors: Mistakes in basic algebraic manipulations, such as combining like terms or distributing, can lead to incorrect solutions. Double-check your steps to avoid these errors.
  5. Not Checking the Solution: After finding a solution set, it's always a good practice to check whether the solutions satisfy the original inequality. This can help identify any mistakes made during the solving process.

Practical Applications of Inequalities

Inequalities are not just abstract mathematical concepts; they have numerous practical applications in various fields. Understanding inequalities can help solve real-world problems in areas such as:

  1. Finance: Inequalities are used to model financial constraints, such as budget limitations or investment returns. For example, an inequality can represent the maximum amount of money a person can spend within a budget.
  2. Engineering: In engineering, inequalities are used to define tolerance levels or safety margins. For instance, an engineer might use an inequality to ensure that a structure can withstand a certain load without exceeding its capacity.
  3. Economics: Economists use inequalities to model supply and demand, price ranges, and other economic constraints. For example, an inequality can represent the range of prices at which a product is profitable to sell.
  4. Computer Science: In computer science, inequalities are used in algorithms for optimization problems, such as finding the shortest path or the most efficient resource allocation.
  5. Statistics: Inequalities are used in statistical analysis to define confidence intervals and test hypotheses. For example, an inequality can represent the range of values within which a population parameter is likely to fall.

Conclusion

In conclusion, the inequality 215 - 2 * 1 - 3 ≤ 15 simplifies to 210 ≤ 15, which is a false statement. Therefore, the solution set is empty, meaning there are no values that satisfy the inequality. None of the provided options match this empty solution set. When solving inequalities, it's crucial to follow a systematic approach, avoid common mistakes, and understand the practical implications of the solutions. Inequalities are a fundamental tool in mathematics with wide-ranging applications in various fields, making their understanding essential for problem-solving and decision-making. Whether you're a student preparing for an exam or a professional applying mathematical concepts in your field, a solid grasp of inequalities will undoubtedly be beneficial. Remember to always double-check your work, understand the implications of your solutions, and appreciate the power of mathematical tools in real-world applications.

By mastering the concepts and techniques discussed in this article, you can confidently tackle inequality problems and apply your knowledge to a variety of practical situations. The ability to solve inequalities is not just a mathematical skill; it's a valuable asset in critical thinking and problem-solving across numerous disciplines.