Hiroto's Equation Error Unveiled Finding The Missing Solution

In the realm of mathematics, solving equations is a fundamental skill, requiring precision and a keen eye for detail. In this article, we delve into the equation $6-4|2 x-8|=-10$, which Hiroto attempted to solve, but only found one solution. Our mission is to dissect his work, pinpoint any errors, and ultimately, uncover the elusive other solution. This exploration will not only help us understand the intricacies of solving absolute value equations but also highlight the importance of meticulousness in mathematical problem-solving.

When tackling equations involving absolute values, it's crucial to remember that the absolute value of a number represents its distance from zero, regardless of direction. This means that the expression inside the absolute value bars can be either positive or negative, leading to two possible scenarios that must be considered. Let's embark on a step-by-step journey through Hiroto's work, scrutinizing each step to identify any potential missteps.

Hiroto's Initial Steps: A Promising Start

Hiroto's initial steps appear to be on the right track. He begins by isolating the absolute value term, a crucial step in solving such equations. Let's examine his first few steps closely:

1. $6-4|2 x-8|=-10$ (Original equation)
2. $-4|2 x-8|=-16$ (Subtracting 6 from both sides)
3. $|2 x-8|=4$ (Dividing both sides by -4)

So far, Hiroto's algebraic manipulations are accurate. He correctly isolates the absolute value expression, setting the stage for the next crucial step: considering both positive and negative cases of the expression inside the absolute value bars. This is where the path diverges, and where we might find the missing solution.

The Critical Bifurcation: Exploring Both Positive and Negative Scenarios

The defining characteristic of absolute value equations lies in their dual nature. The expression within the absolute value bars can be either positive or negative, yet yield the same absolute value. This necessitates splitting the equation into two distinct cases:

• Case 1: The positive scenario

  When the expression inside the absolute value, $(2x-8)$, is positive or zero, the absolute value remains unchanged. This leads to the equation:

  $2x - 8 = 4$

  This is precisely the equation Hiroto proceeded with, leading him to the solution $x = 6$. However, this is only half the story. Let's delve into the other possibility.

• Case 2: The negative scenario

  When the expression $(2x-8)$ is negative, the absolute value negates it, effectively changing its sign. This gives us the equation:

  $-(2x - 8) = 4$

  This is the crucial step Hiroto missed, and it holds the key to unlocking the other solution. By neglecting this case, he inadvertently overlooked a significant part of the solution set.

Unveiling the Missing Solution: Solving the Negative Scenario

To find the missing solution, we must now tackle the equation derived from the negative scenario:

$-(2x - 8) = 4$

Let's break down the steps:

1. Distribute the negative sign: $-2x + 8 = 4$
2. Subtract 8 from both sides: $-2x = -4$
3. Divide both sides by -2: $x = 2$

And there it is! The missing solution is $x = 2$. This solution arises from considering the case where the expression inside the absolute value is negative, a crucial step that Hiroto omitted in his attempt.

Hiroto's Oversight: The Importance of Considering All Cases

Hiroto's mistake serves as a valuable lesson in the meticulousness required for mathematical problem-solving. He correctly isolated the absolute value and solved for one case, but failed to account for the negative scenario. This oversight highlights the importance of understanding the fundamental properties of mathematical concepts, in this case, the absolute value.

When dealing with absolute values, always remember to consider both possibilities: the expression inside the absolute value bars can be either positive or negative. Failing to do so can lead to incomplete solutions and a skewed understanding of the problem.

The Complete Solution Set: A Comprehensive Understanding

By considering both the positive and negative scenarios, we arrive at the complete solution set for the equation $6-4|2 x-8|=-10$. The solutions are $x = 6$ and $x = 2$. These values satisfy the original equation, confirming the necessity of exploring all possible cases when dealing with absolute values.

This exercise underscores the significance of a thorough and comprehensive approach to problem-solving in mathematics. It's not enough to find one solution; we must strive to uncover all possible solutions to gain a complete understanding of the problem.

Having solved the equation completely, we now know that the other solution, which Hiroto missed, is $x = 2$. Therefore, the correct answer among the given options is:

C. 2

This exploration of Hiroto's equation-solving journey offers several valuable insights into tackling absolute value equations:

• Isolate the absolute value: The first step in solving an absolute value equation is to isolate the absolute value term. This involves performing algebraic manipulations to get the absolute value expression by itself on one side of the equation.
• Consider both positive and negative cases: The defining characteristic of absolute value equations is the need to consider both positive and negative scenarios for the expression inside the absolute value bars. This leads to two separate equations that must be solved.
• Solve each case independently: Once you've split the equation into two cases, solve each one independently using standard algebraic techniques.
• Check your solutions: After finding potential solutions, it's always a good practice to plug them back into the original equation to verify their validity. This helps ensure that you haven't introduced any extraneous solutions.
• Meticulousness is key: Mathematical problem-solving demands precision and attention to detail. Overlooking a crucial step, such as considering both positive and negative cases in an absolute value equation, can lead to incomplete or incorrect solutions.

By adhering to these principles, you can confidently navigate the world of absolute value equations and unlock their solutions with accuracy and precision. Remember, the key is to embrace a comprehensive and methodical approach, leaving no stone unturned in your quest for mathematical understanding.

To truly master absolute value equations, it's essential to have a solid conceptual understanding of absolute value itself. The absolute value of a number is its distance from zero on the number line. This distance is always non-negative, regardless of whether the number itself is positive or negative.

For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5. This is because both 5 and -5 are 5 units away from zero. The absolute value is denoted by vertical bars, so we write $|5| = 5$ and $|-5| = 5$.

This understanding of absolute value as a distance is crucial for solving equations. When we see an equation like $|2x - 8| = 4$, we're essentially asking: