Emilia's Inequality Error Unveiling The Distributive Property Mistake
In the realm of mathematics, solving inequalities is a fundamental skill. But, like any mathematical process, it's crucial to execute each step with precision. Today, we'll dissect a particular inequality problem solved by Emilia, highlighting a critical error in her steps. Let's delve into the step-by-step solution and pinpoint where the misstep occurred. Emilia's attempt to solve the inequality provides a valuable learning opportunity to reinforce the correct procedures and properties involved in inequality manipulation. Understanding these properties is essential for accurate problem-solving and for building a solid foundation in algebra. Through careful examination and explanation, we aim to clarify the concepts and prevent similar errors in the future. This exercise not only helps in identifying mistakes but also in strengthening our understanding of the underlying principles of inequalities. It’s a crucial part of mastering mathematical problem-solving, where attention to detail and correct application of properties are paramount. Let's begin by reviewing the given steps and then focus on the property that should have been applied correctly.
Emilia embarked on solving the inequality, and her journey can be traced through these steps:
- Step 1:
- Step 2:
- Step 3:
- Step 4:
Our mission is to identify the property that justifies the transition from Step 2 to Step 3. However, before we jump to conclusions, let's analyze the steps closely. This careful examination will allow us to understand the logical progression and pinpoint any errors. The process of solving inequalities involves applying various algebraic properties, similar to solving equations, but with certain key differences. Recognizing and correctly applying these properties is vital for arriving at the correct solution. Each step must be justified by a valid mathematical operation or property. By scrutinizing each step, we not only find the error but also reinforce our understanding of these fundamental mathematical principles. This analytical approach is crucial for developing strong problem-solving skills in mathematics. We'll take a closer look at the properties that govern inequality manipulation to ensure clarity.
The glaring error lies in the transition from Step 1 to Step 2. In Step 1, we have:
To move to the next step, the distributive property should be applied. This means multiplying -2 by both terms inside the parentheses: -2 * x and -2 * -3. The correct application of the distributive property is a cornerstone of algebraic manipulation. It ensures that each term within the parentheses is correctly accounted for in the equation or inequality. A misapplication here can lead to significant errors in the final solution. It's a fundamental principle that must be mastered to solve algebraic problems accurately. Understanding how the distributive property interacts with negative numbers is particularly important, as this is a common source of mistakes. Let’s perform this step correctly to see how the inequality should have been transformed.
When we correctly distribute the -2, we get:
Now, comparing this correct result with Emilia's Step 2, which states "2x - 6 > x + 9," the mistake becomes evident. Emilia incorrectly distributed the -2, leading to the wrong signs for both the 'x' term and the constant term. This error highlights the critical importance of paying close attention to signs when performing algebraic operations. A simple sign mistake can completely alter the outcome of the problem. The impact of this initial error will propagate through the rest of the steps, resulting in an incorrect solution. Therefore, identifying and correcting this mistake is essential for understanding the correct problem-solving approach. The negative sign plays a crucial role in determining the correct result, and its misapplication is a common error in algebra. Let's now analyze why the correct distribution is crucial for solving the inequality accurately.
Starting with the corrected Step 2, we have:
The next step involves isolating the 'x' terms on one side of the inequality and the constants on the other. To achieve this, we can add 2x to both sides of the inequality. This operation is justified by the addition property of inequality, which states that adding the same value to both sides of an inequality preserves the inequality. This property is a fundamental tool in solving inequalities, allowing us to manipulate the expression while maintaining its validity. By adding 2x to both sides, we move closer to isolating the variable 'x' and simplifying the inequality. The addition property, like other properties of inequality, is crucial for maintaining the correct relationship between the two sides of the inequality throughout the solution process. Let's execute this step and see what the resulting inequality looks like.
Adding 2x to both sides, we get:
Now, we need to isolate the 'x' term further. To do this, we subtract 9 from both sides. This again is justified by the subtraction property of inequality, which, similar to the addition property, allows us to subtract the same value from both sides without changing the inequality's direction. The subtraction property is another crucial tool in our algebraic arsenal for solving inequalities. By carefully applying this property, we can systematically isolate the variable we're trying to solve for. Each step we take brings us closer to the final solution, and the correct application of these properties ensures the accuracy of our results. Let’s perform the subtraction and simplify the inequality further.
Subtracting 9 from both sides, we have:
Finally, to solve for 'x', we divide both sides by 3. However, there's a crucial rule to remember when dividing (or multiplying) an inequality by a negative number: we must flip the inequality sign. Since we are dividing by a positive 3, we do not need to flip the sign in this case. This rule is a critical aspect of solving inequalities and stems from the way negative numbers interact with inequalities. Failing to flip the sign when dividing by a negative number will result in an incorrect solution. This step highlights the nuanced differences between solving equations and solving inequalities. Let’s complete this final step and arrive at the correct solution for 'x'.
Dividing both sides by 3, we obtain:
This can also be written as:
Going back to the original question, we need to identify the property that justifies the work between Step 2 and Step 3 in Emilia's solution. However, as we've established, there's an error in Step 2 itself. The correct property that should have been applied between Step 1 and a corrected Step 2 is the distributive property. The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms within parentheses. It’s a cornerstone of algebraic simplification and manipulation. Understanding and applying the distributive property correctly is crucial for success in algebra. It ensures that expressions are correctly expanded and simplified, leading to accurate solutions. Let's delve deeper into why this property is so important and how it works in detail.
The distributive property states that for any numbers a, b, and c:
In our case, this translates to multiplying -2 by both 'x' and '-3' inside the parentheses. The failure to apply this property correctly led to the error in Emilia's solution. The distributive property is not just a mathematical rule; it's a fundamental principle that governs how we manipulate algebraic expressions. Its correct application is vital for simplifying complex expressions and for accurately solving equations and inequalities. Mastery of this property is a key stepping stone in developing strong algebraic skills. Let's reiterate why applying the distributive property correctly is essential for solving this inequality and similar problems.
The distributive property is more than just a rule; it's a critical tool for accurate algebraic manipulation. In the context of inequalities and equations, its correct application is paramount to arriving at the correct solution. When a term is multiplied by an expression in parentheses, each term inside the parentheses must be multiplied by the term outside. Ignoring or misapplying this rule leads to incorrect expressions, and subsequently, incorrect solutions. The distributive property is particularly important when dealing with negative numbers, as the signs must be carefully managed. A negative sign outside the parentheses changes the sign of each term inside when the distributive property is applied. This sensitivity to signs makes it crucial to pay close attention when distributing. Let's further explore the implications of the distributive property in various algebraic contexts.
In Emilia's case, the incorrect distribution of -2 resulted in a flawed Step 2, which then propagated through the rest of her solution. This highlights the importance of checking each step carefully, especially when applying fundamental properties like the distributive property. A small error early in the process can have significant consequences for the final answer. This emphasizes the need for a meticulous and methodical approach to problem-solving in mathematics. Each step should be scrutinized to ensure accuracy and adherence to the correct mathematical principles. Let’s revisit the corrected steps to reinforce the importance of the distributive property in obtaining the correct solution.
In conclusion, the error in Emilia's solution lies in the incorrect application of the distributive property in the initial steps. The correct application of this property, along with a careful consideration of the rules for manipulating inequalities (especially when multiplying or dividing by negative numbers), is crucial for arriving at the correct solution. This exercise underscores the importance of mastering the fundamental properties of algebra. These properties are the building blocks of more advanced mathematical concepts. A strong understanding of these fundamentals is essential for success in mathematics. By carefully applying these properties and paying attention to detail, we can avoid common pitfalls and solve mathematical problems accurately. Let's strive to solidify our understanding of these fundamentals and apply them with confidence in our future mathematical endeavors.
By identifying and correcting the error, we've not only solved the problem but also reinforced our understanding of the distributive property and its significance in solving inequalities. Remember, in mathematics, precision and a thorough understanding of fundamental principles are key to success. This detailed analysis serves as a reminder to always double-check each step and ensure that the correct properties are being applied.
