First Step To Solve (m-2)/6 < -1 An Inequality Guide

JU07/21/2025 08, 2025 by THE IDEN 53 views

Unveiling the First Step in Solving the Inequality (m-2)/6 < -1

Introduction

Solving inequalities is a fundamental concept in mathematics, and it's essential to understand the steps involved in finding the solution set. When confronted with an inequality such as (m-2)/6 < -1, knowing the initial move is crucial for simplification and ultimately determining the range of values for the variable 'm' that satisfy the condition. This article will provide a detailed exploration of the first step in solving this inequality, offering insights into the underlying principles and practical application of the concept.

Understanding Inequalities

Before diving into the specifics of solving (m-2)/6 < -1, let's first establish a clear understanding of what inequalities are and how they differ from equations. In mathematics, an inequality is a statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike equations, which assert the equality of two expressions, inequalities indicate a range of possible values that satisfy the given condition. When solving inequalities, our goal is to isolate the variable on one side of the inequality symbol to determine the set of values that make the inequality true. This process often involves performing operations on both sides of the inequality, similar to solving equations, but with a crucial difference: multiplying or dividing by a negative number requires flipping the direction of the inequality symbol. Inequalities play a vital role in various mathematical and real-world applications, from determining the feasible region in linear programming to modeling constraints in optimization problems. They provide a powerful tool for expressing relationships where exact equality is not necessary or possible, allowing us to analyze and solve problems involving a range of solutions.

Identifying the First Step

In this particular inequality, (m-2)/6 < -1, the first step is to eliminate the fraction. The fraction can be eliminated by multiplying both sides of the inequality by the denominator, which is 6. The primary goal in solving any algebraic inequality or equation is to isolate the variable on one side. To do this effectively, we must systematically undo the operations that are being performed on the variable. In this case, 'm' is being subtracted by 2, and the result is being divided by 6. The order of operations (PEMDAS/BODMAS) dictates that we address division before subtraction when working backward to isolate the variable. Therefore, the logical first step is to counteract the division by 6. Multiplying both sides of the inequality by 6 maintains the balance of the inequality while eliminating the fraction, making the expression simpler to work with. This approach aligns with the fundamental principle of maintaining equality or inequality by performing the same operation on both sides. By clearing the fraction in the initial step, we pave the way for a more straightforward solution process, reducing the complexity of the inequality and bringing us closer to isolating 'm'. This strategic move demonstrates an understanding of algebraic manipulation and sets the stage for subsequent steps in solving the inequality.

Step-by-Step Explanation

To solve the inequality (m-2)/6 < -1, the initial step involves multiplying both sides of the inequality by 6. Let's break this down step by step:

Write down the original inequality: (m-2)/6 < -1
Multiply both sides by 6: 6 * [(m-2)/6] < 6 * (-1)
Simplify both sides: On the left side, the multiplication by 6 cancels out the division by 6, leaving (m-2). On the right side, 6 multiplied by -1 equals -6. So, the simplified inequality becomes: m - 2 < -6 This step is crucial because it eliminates the fraction, making the inequality easier to manipulate. By multiplying both sides by 6, we maintain the balance of the inequality while transforming it into a more manageable form. Now, we have a simple inequality where the variable 'm' is only being subjected to subtraction. The next steps will involve isolating 'm' by addressing the subtraction operation. This systematic approach of eliminating fractions and simplifying the expression is a fundamental technique in solving algebraic inequalities and equations. It allows us to gradually isolate the variable and determine the solution set. The importance of this step cannot be overstated, as it lays the foundation for the subsequent steps in the solution process. By carefully executing this multiplication, we ensure that the inequality remains balanced and that we are progressing towards the correct solution.

Why This Is the Correct First Step

The reason why multiplying both sides by 6 is the correct first step lies in the order of operations and the principles of algebraic manipulation. In algebra, our goal when solving for a variable is to isolate it on one side of the equation or inequality. This involves systematically undoing the operations that are being performed on the variable, working in reverse order of the typical order of operations (PEMDAS/BODMAS). In the inequality (m-2)/6 < -1, the variable 'm' is first subtracted by 2, and then the result is divided by 6. To isolate 'm', we must reverse these operations. According to the order of operations, division and multiplication are performed before addition and subtraction. Therefore, to undo the operations affecting 'm', we should first address the division by 6. This is achieved by multiplying both sides of the inequality by 6. This action effectively cancels out the division by 6 on the left side, simplifying the expression and bringing us closer to isolating 'm'. Moreover, multiplying both sides of an inequality by a positive number maintains the direction of the inequality, which is crucial for preserving the integrity of the solution. If we were to address the subtraction first, we would need to add 2 to both sides, resulting in (m-2)/6 + 2 < -1 + 2. While this is a valid operation, it doesn't directly eliminate the fraction, making the subsequent steps more complex. By multiplying by 6 first, we streamline the process and avoid unnecessary complications. This approach demonstrates a strategic understanding of algebraic principles and efficiently guides us towards the solution.

Common Mistakes to Avoid

When solving inequalities, there are several common mistakes that students often make. One of the most frequent errors is forgetting to flip the direction of the inequality sign when multiplying or dividing both sides by a negative number. This mistake can lead to an incorrect solution set. Another common mistake is incorrectly applying the order of operations. For instance, in the inequality (m-2)/6 < -1, some students might mistakenly add 2 to both sides before multiplying by 6. This approach, while not inherently wrong, adds an unnecessary step and can complicate the process. It's crucial to address the division by 6 first to simplify the inequality efficiently. Another error arises when students distribute incorrectly. If there were a coefficient outside the parentheses in the numerator, such as 2(m-2)/6 < -1, it's essential to distribute the 2 across both terms inside the parentheses before proceeding with other operations. Failing to do so can lead to an incorrect simplification. Additionally, students might make mistakes when combining like terms or simplifying expressions on either side of the inequality. Careful attention to detail and a systematic approach can help prevent these errors. Always double-check your work and ensure that each step is logically sound. Practicing with a variety of inequality problems can also help solidify your understanding and reduce the likelihood of making these common mistakes. By being aware of these pitfalls and taking steps to avoid them, you can confidently solve inequalities and arrive at the correct solution.

Next Steps in Solving the Inequality

After multiplying both sides of the inequality (m-2)/6 < -1 by 6, we arrive at the simplified inequality m - 2 < -6. The next step is to isolate the variable 'm' by undoing the subtraction of 2. To achieve this, we add 2 to both sides of the inequality. This operation maintains the balance of the inequality and moves us closer to the solution. Adding 2 to both sides, we get: m - 2 + 2 < -6 + 2 Simplifying both sides, we have: m < -4 This result tells us that 'm' is less than -4. To fully understand the solution, it's helpful to represent it graphically on a number line. Draw a number line and mark -4 on it. Since the inequality is strictly less than (-4), we use an open circle at -4 to indicate that -4 itself is not included in the solution set. Then, shade the region to the left of -4, representing all the values less than -4. This visual representation provides a clear understanding of the range of values that satisfy the inequality. In interval notation, the solution is expressed as (-∞, -4). This notation indicates that the solution includes all real numbers from negative infinity up to, but not including, -4. Understanding how to express the solution in different forms – algebraically, graphically, and in interval notation – is crucial for a comprehensive understanding of inequality solutions. By following these steps, we can confidently solve the inequality and accurately represent the solution set.

Real-World Applications

Inequalities, like the one we solved, have numerous real-world applications across various fields. They are essential tools for modeling situations where exact equality is not required or possible, but rather a range of values is acceptable or desirable. In economics, inequalities are used to represent budget constraints, production possibilities, and supply and demand relationships. For example, a consumer's budget constraint might be expressed as an inequality, indicating that the total spending on goods and services must be less than or equal to the consumer's income. In engineering, inequalities are used to define tolerance limits, safety margins, and performance specifications. For instance, a bridge might be designed to withstand a certain load, with the actual load needing to be less than or equal to the design capacity. In computer science, inequalities are used in algorithm analysis to express time and space complexity, and in optimization problems to define constraints on resources. In everyday life, inequalities are used in various decision-making processes. For example, when planning a road trip, you might use an inequality to determine the maximum distance you can travel given your budget for fuel and accommodation. Similarly, when following a recipe, you might use inequalities to adjust the quantities of ingredients based on the number of servings you want to make. Inequalities also play a crucial role in statistics and data analysis, where they are used to define confidence intervals, hypothesis tests, and probability distributions. Understanding how to solve and interpret inequalities is therefore a valuable skill that extends far beyond the classroom, enabling us to model and solve problems in a wide range of practical contexts. The ability to translate real-world scenarios into mathematical inequalities and then solve them is a powerful tool for informed decision-making and problem-solving.

Conclusion

In conclusion, solving the inequality (m-2)/6 < -1 begins with multiplying both sides by 6. This step effectively eliminates the fraction and simplifies the inequality, setting the stage for further steps to isolate the variable 'm'. Understanding the underlying principles of algebraic manipulation and the order of operations is crucial for solving inequalities accurately. By avoiding common mistakes and following a systematic approach, we can confidently determine the solution set and represent it in various forms. Inequalities are a fundamental concept in mathematics with widespread applications in real-world scenarios, making it essential to grasp the techniques involved in solving them. Mastering this skill empowers us to tackle a wide range of problems and make informed decisions in various contexts. The journey of solving inequalities is not just about finding the right answer; it's about developing a logical and analytical mindset that can be applied to diverse challenges. By understanding the principles and practicing consistently, we can become proficient in solving inequalities and harness their power to model and solve real-world problems.