Yuto And Hila's Inequality Challenge Unveiling The Solution

In the fascinating world of mathematics, solving inequalities is a fundamental skill. Inequalities, unlike equations, deal with relationships where one value is greater than, less than, or not equal to another. Today, we delve into a mathematical puzzle involving two students, Yuto and Hila, who embarked on a journey to solve the same inequality. Let's analyze their approaches, unravel their steps, and uncover any potential errors or ingenious solutions.

The Inequality Challenge

The inequality at the heart of our investigation is:

$-\frac{2}{3}x + 9 > 7$

This inequality presents a linear relationship where we seek to find the values of 'x' that satisfy the condition. Both Yuto and Hila attempted to solve this, and their work is meticulously documented in the table below:

Decoding Yuto's Method: A Step-by-Step Analysis

Yuto's approach to solving the inequality is a classic application of algebraic principles. Yuto started with the original inequality, $-\frac{2}{3}x + 9 > 7$. The initial step involved isolating the term containing 'x'. To achieve this, Yuto subtracted 9 from both sides of the inequality. This operation maintains the balance of the inequality, similar to how we manipulate equations. The result was a simplified inequality: $-\frac{2}{3}x > -2$. This step is crucial as it brings us closer to isolating 'x'. To isolate 'x' completely, Yuto multiplied both sides of the inequality by $-\frac{3}{2}$. This is where a critical rule comes into play: when multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed. Yuto correctly applied this rule, changing the '>' sign to '<'. The multiplication yields $x < 3$, which means that all values of 'x' less than 3 satisfy the original inequality. Yuto's solution suggests that any number smaller than 3, when substituted for 'x' in the original inequality, will make the statement true. For example, if we take x = 0, we get $-\frac{2}{3}(0) + 9 > 7$, which simplifies to 9 > 7, a true statement. Similarly, if we take x = 2, we have $-\frac{2}{3}(2) + 9 > 7$, which simplifies to approximately 7.67 > 7, also true. Yuto's meticulous application of algebraic principles, including the crucial sign reversal, showcases a strong understanding of inequality manipulation.

Hila's Inequality Expedition: Navigating the Solution Path

Hila's journey towards solving the inequality mirrors Yuto's in the initial stages. Hila began with the same inequality, $-\frac{2}{3}x + 9 > 7$, and mirrored Yuto's first step by subtracting 9 from both sides. This subtraction, a fundamental algebraic operation, maintains the balance and simplifies the inequality to $-\frac{2}{3}x > -2$. At this point, the paths diverge slightly, but the core principle remains the same: isolating 'x'. To isolate 'x', Hila, like Yuto, multiplied both sides of the inequality by $-\frac{3}{2}$. This step is a pivotal moment in the solution process, as it involves multiplying by a negative number. The critical rule here is that multiplying or dividing an inequality by a negative number necessitates flipping the direction of the inequality sign. Here's where the key difference lies between Hila's and Yuto's work. Hila multiplied by $-\frac{3}{2}$ but did not reverse the inequality sign, leading to an incorrect conclusion. The incorrect multiplication resulted in $x > 3$, which suggests that all values of 'x' greater than 3 satisfy the original inequality. This is in direct contrast to Yuto's solution. To illustrate the error, let's test x = 4 in the original inequality: $-\frac{2}{3}(4) + 9 > 7$ simplifies to approximately 6.33 > 7, which is false. This demonstrates that Hila's solution is not universally valid and highlights the significance of the sign reversal rule. Hila's work, while demonstrating an understanding of initial algebraic steps, faltered at the crucial application of the negative multiplication rule.

Unveiling the Discrepancy: The Sign Reversal Saga

The difference in the solutions obtained by Yuto and Hila stems from a single, yet critical, step: the handling of the inequality sign when multiplying by a negative number. This is a common pitfall in solving inequalities, and understanding the underlying reason for this rule is essential. The rule dictates that when an inequality is multiplied or divided by a negative number, the direction of the inequality sign must be flipped. This is because multiplying by a negative number effectively reverses the number line. For example, if 2 < 4, multiplying both sides by -1 gives -2 > -4. The relative order of the numbers is reversed. In the context of our inequality, multiplying $-\frac{2}{3}x > -2$ by $-\frac{3}{2}$ requires flipping the '>' sign to '<', as Yuto correctly did. Hila's failure to do so led to an incorrect solution set. This highlights the importance of not just memorizing the rule, but understanding the mathematical reasoning behind it. The sign reversal isn't an arbitrary step; it's a consequence of how negative numbers interact with inequalities.

Validating the Solutions: Proof in the Numbers

To definitively determine the correct solution, we can test values in the original inequality. Yuto's solution suggests that $x < 3$. Let's test x = 2:

$-\frac{2}{3}(2) + 9 > 7$

$-\frac{4}{3} + 9 > 7$

$7.67 > 7$ (approximately), which is true.

Now, let's test x = 4 (which should not satisfy the inequality according to Yuto's solution):

$-\frac{2}{3}(4) + 9 > 7$

$-\frac{8}{3} + 9 > 7$

$6.33 > 7$ (approximately), which is false.

This confirms Yuto's solution. Testing values is a powerful method for validating inequality solutions. For Hila's solution ($x > 3$), let's test x = 4 (which Hila's solution suggests should work):

$-\frac{2}{3}(4) + 9 > 7$

$-\frac{8}{3} + 9 > 7$

$6.33 > 7$ (approximately), which is false.

This demonstrates that Hila's solution is incorrect. The validation process underscores the importance of double-checking solutions, especially in inequalities where sign errors can easily occur.

Lessons Learned: Mastering Inequality Manipulation

The case of Yuto and Hila provides valuable insights into solving inequalities. The key takeaway is the critical importance of the sign reversal rule when multiplying or dividing by a negative number. This rule is not just a procedural step; it's a fundamental aspect of how inequalities work. Understanding the mathematical reasoning behind it, rather than simply memorizing it, leads to a deeper understanding and fewer errors. Additionally, the process of validating solutions by testing values is an invaluable tool. It not only confirms the correctness of the solution but also reinforces the understanding of what the inequality represents. Furthermore, this scenario highlights the importance of careful attention to detail in mathematical problem-solving. A single missed sign can lead to an entirely incorrect solution. By dissecting the approaches of Yuto and Hila, we gain a clearer perspective on the nuances of inequality manipulation and the significance of each step in the solution process. Finally, this exploration emphasizes the value of peer review and collaborative learning in mathematics. Discussing different approaches and identifying errors together can enhance understanding and prevent common mistakes.

Conclusion: The Triumph of Correct Application

In the end, Yuto's solution, $x < 3$, stands as the correct answer. His meticulous application of algebraic principles, including the crucial sign reversal, demonstrates a strong grasp of inequality manipulation. Hila's attempt, while displaying an understanding of initial steps, stumbled at the pivotal moment of multiplying by a negative number, highlighting the importance of this rule. This mathematical exploration serves as a reminder that precision and attention to detail are paramount in mathematics. The story of Yuto and Hila's inequality adventure is not just about finding the right answer; it's about the journey of understanding the underlying principles and avoiding common pitfalls. By learning from their experiences, we can all become more proficient inequality solvers. Ultimately, the triumph lies not just in arriving at the correct solution, but in the process of learning and understanding the mathematical concepts involved.