Solving Absolute Value Equations A Comprehensive Guide To |2x-6|+6=74
In the realm of mathematics, absolute value equations often pose a unique challenge, demanding a keen understanding of the absolute value concept itself. The absolute value of a number represents its distance from zero on the number line, irrespective of direction. This inherent property gives rise to two possible scenarios when dealing with absolute value equations, as the expression inside the absolute value bars can be either positive or negative, yet yield the same absolute value. This article delves into a detailed exploration of how to solve the equation |2x - 6| + 6 = 74, providing a step-by-step approach suitable for students and math enthusiasts alike. Our primary focus will be on unraveling the complexities of absolute value and applying the appropriate techniques to find the possible values of the variable x. Let's embark on this mathematical journey, where clarity and precision are our guiding principles. Absolute value is a fundamental concept in mathematics, and mastering it is crucial for success in algebra and beyond. The ability to solve absolute value equations opens doors to more advanced mathematical concepts and applications, making it an invaluable skill for any aspiring mathematician or student. Understanding the nuances of absolute value also enhances problem-solving abilities in various real-world scenarios, where considering both positive and negative possibilities is essential. So, as we dive into this equation, remember that we are not just solving a problem; we are building a foundation for future mathematical endeavors.
Step 1: Isolating the Absolute Value Expression
Before we can tackle the absolute value, our initial task is to isolate the absolute value expression on one side of the equation. This involves performing algebraic manipulations to ensure that the |2x - 6| term stands alone, free from any additional constants or coefficients. In our given equation, |2x - 6| + 6 = 74, we observe that the absolute value expression is accompanied by a '+ 6' term. To isolate |2x - 6|, we need to eliminate this '+ 6'. The fundamental principle of algebraic manipulation dictates that any operation performed on one side of the equation must also be performed on the other side to maintain the equality. Therefore, we subtract 6 from both sides of the equation. This step is crucial as it simplifies the equation and sets the stage for addressing the absolute value directly. Subtracting 6 from both sides gives us: |2x - 6| + 6 - 6 = 74 - 6, which simplifies to |2x - 6| = 68. Now, with the absolute value expression isolated, we can proceed to the next phase of the solution, where we consider the two possible cases that arise from the nature of absolute value. Remember, isolating the absolute value expression is a standard first step in solving such equations, and mastering this step is key to tackling more complex problems involving absolute values.
Step 2: Considering the Two Cases
With the absolute value expression now isolated, we arrive at the pivotal step in solving absolute value equations: considering the two possible cases. The essence of absolute value lies in its dual nature – a number within the absolute value bars can be either positive or negative, yet its absolute value remains positive. This stems from the definition of absolute value as the distance from zero, which is always a non-negative quantity. In our equation, |2x - 6| = 68, this implies that the expression 2x - 6 can be either 68 or -68, as both values have an absolute value of 68. This is where the equation bifurcates into two separate paths, each requiring individual attention. Case 1: The Positive Case. We begin by assuming that the expression inside the absolute value bars, 2x - 6, is equal to the positive value, 68. This gives us the equation 2x - 6 = 68. This equation represents one possible scenario, where the quantity within the absolute value is positive. Case 2: The Negative Case. The second possibility arises when the expression 2x - 6 is equal to the negative value, -68. This leads us to the equation 2x - 6 = -68. This case accounts for the scenario where the quantity within the absolute value is negative, but its absolute value is still 68. By acknowledging and addressing these two cases separately, we ensure that we capture all possible solutions for x that satisfy the original equation. The next step involves solving each of these equations independently to determine the potential values of x. Understanding the significance of these two cases is fundamental to mastering absolute value equations.
Step 3: Solving for x in Case 1 (2x - 6 = 68)
Having established the two cases, we now turn our attention to solving for x in each scenario. Case 1, where 2x - 6 = 68, represents the situation where the expression inside the absolute value is positive. To solve this equation, we employ the standard algebraic techniques of isolating the variable x. The first step involves undoing the subtraction by adding 6 to both sides of the equation. This maintains the balance of the equation and moves us closer to isolating the term containing x. Adding 6 to both sides of 2x - 6 = 68 gives us 2x - 6 + 6 = 68 + 6, which simplifies to 2x = 74. Now, we have the equation 2x = 74. The next step in isolating x is to eliminate the coefficient 2. This is achieved by dividing both sides of the equation by 2. Division is the inverse operation of multiplication, and performing this operation on both sides ensures that the equation remains balanced. Dividing both sides of 2x = 74 by 2 yields 2x / 2 = 74 / 2, which simplifies to x = 37. Therefore, in Case 1, we find that x = 37 is a potential solution to the original equation. However, this is just one possibility, and we must also consider Case 2 to determine all possible values of x. The systematic approach of adding and dividing, while ensuring that the equation remains balanced, is a cornerstone of algebraic problem-solving.
Step 4: Solving for x in Case 2 (2x - 6 = -68)
Having successfully solved for x in Case 1, we now shift our focus to Case 2, where 2x - 6 = -68. This case addresses the scenario where the expression inside the absolute value bars is negative. Similar to Case 1, our objective is to isolate x using algebraic manipulations while maintaining the equality of the equation. We begin by undoing the subtraction of 6. To do this, we add 6 to both sides of the equation. Adding 6 to both sides of 2x - 6 = -68 gives us 2x - 6 + 6 = -68 + 6, which simplifies to 2x = -62. This step is crucial as it brings us closer to isolating the term containing x. Now that we have the equation 2x = -62, the next step is to eliminate the coefficient 2. We achieve this by dividing both sides of the equation by 2. Dividing both sides of 2x = -62 by 2 yields 2x / 2 = -62 / 2, which simplifies to x = -31. Therefore, in Case 2, we find that x = -31 is another potential solution to the original equation. We now have two potential solutions: x = 37 from Case 1 and x = -31 from Case 2. It is important to remember that these are potential solutions and should be verified in the original equation to ensure their validity. The process of solving for x in Case 2 reinforces the importance of careful manipulation and attention to signs in algebraic equations. Each step must be executed with precision to arrive at the correct solution.
Step 5: Verifying the Solutions
After finding potential solutions for an equation, especially those involving absolute values, it is paramount to verify these solutions. This step ensures that the values we've obtained for x indeed satisfy the original equation and are not extraneous solutions introduced during the solving process. Verification is a crucial safeguard against errors and a testament to the accuracy of our solution. We have two potential solutions to verify: x = 37 and x = -31. To verify these solutions, we substitute each value back into the original equation, |2x - 6| + 6 = 74, and check if the equation holds true. Verification for x = 37. Substituting x = 37 into the original equation gives us |2(37) - 6| + 6 = 74. Simplifying inside the absolute value, we get |74 - 6| + 6 = 74, which further simplifies to |68| + 6 = 74. Since the absolute value of 68 is 68, we have 68 + 6 = 74, which is indeed true. Therefore, x = 37 is a valid solution. Verification for x = -31. Substituting x = -31 into the original equation gives us |2(-31) - 6| + 6 = 74. Simplifying inside the absolute value, we get |-62 - 6| + 6 = 74, which further simplifies to |-68| + 6 = 74. Since the absolute value of -68 is 68, we have 68 + 6 = 74, which is also true. Therefore, x = -31 is a valid solution. By successfully verifying both potential solutions, we can confidently state that x = 37 and x = -31 are the solutions to the equation |2x - 6| + 6 = 74. This verification process underscores the importance of rigor in mathematical problem-solving and ensures that our solutions are accurate and reliable.
Final Answer and Conclusion
In conclusion, after meticulously solving the absolute value equation |2x - 6| + 6 = 74, we have arrived at two valid solutions: x = 37 and x = -31. These solutions were obtained by first isolating the absolute value expression, then considering the two possible cases arising from the nature of absolute value—where the expression inside the absolute value bars can be either positive or negative. Each case led to a separate linear equation, which we solved using standard algebraic techniques. Finally, we verified both solutions by substituting them back into the original equation, confirming their validity. The final answer is that the possible values of x are 37 and -31. This comprehensive approach to solving absolute value equations highlights the importance of understanding the underlying concepts, applying algebraic techniques with precision, and verifying solutions to ensure accuracy. Absolute value equations, while seemingly complex, can be systematically solved by breaking them down into manageable steps. The key takeaway from this exercise is the ability to recognize the dual nature of absolute value and to address each possibility with careful consideration. Mastering the techniques demonstrated in this article will not only enable you to solve similar equations but also enhance your problem-solving skills in mathematics more broadly. Remember, practice and a clear understanding of fundamental concepts are the cornerstones of mathematical proficiency. We trust this detailed exploration has provided you with valuable insights and the confidence to tackle absolute value equations effectively.
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