Solve The Absolute Value Equation |3x| - 4 = 2x - 4.

In the realm of mathematics, absolute value equations often present a unique challenge due to the nature of the absolute value function. The absolute value of a number is its distance from zero, meaning it is always non-negative. This characteristic leads to two possible scenarios when solving equations involving absolute values, as a number inside the absolute value can be either positive or negative while yielding the same absolute value. To accurately solve an absolute value equation, it is crucial to consider both possibilities and determine the solutions that satisfy the original equation. This article will delve into the step-by-step solution of the absolute value equation |3x| - 4 = 2x - 4, providing a clear and comprehensive guide for readers to understand and apply the methodology. We will explore the fundamental principles behind absolute value equations, the techniques used to isolate the absolute value term, and the process of splitting the equation into two separate cases to account for both positive and negative scenarios. By the end of this discussion, readers will have a solid grasp of how to approach and solve similar equations, ensuring they can confidently tackle such problems in their mathematical pursuits. This comprehensive understanding is not only essential for academic success but also for real-world applications where absolute values are used to model distances, tolerances, and other non-negative quantities. The ability to solve absolute value equations effectively is a cornerstone of mathematical literacy, empowering individuals to analyze and interpret quantitative information with precision and clarity.

Problem Statement: Solving |3x| - 4 = 2x - 4

The task at hand is to solve the absolute value equation |3x| - 4 = 2x - 4 for the variable x. This equation involves the absolute value of 3x, which means we need to consider two separate cases: one where 3x is non-negative (3x ≥ 0) and another where 3x is negative (3x < 0). The absolute value function, denoted by | |, returns the magnitude of a number, disregarding its sign. Therefore, |3x| will be equal to 3x if 3x is non-negative and -3x if 3x is negative. This dual nature of the absolute value function is what necessitates the splitting of the equation into two distinct cases. By addressing each case separately, we can systematically eliminate the absolute value and obtain linear equations that are more straightforward to solve. The solutions obtained from each case must then be verified in the original equation to ensure they are not extraneous, as the process of squaring or manipulating equations can sometimes introduce solutions that do not satisfy the original problem. This rigorous approach ensures that the final answers are accurate and valid, providing a complete and reliable solution to the given absolute value equation. The careful consideration of both positive and negative scenarios is paramount in solving absolute value equations, and this step-by-step process will guide us through to the correct solutions.

Step 1: Isolate the Absolute Value Term

Before diving into the different cases, the initial step in solving the equation |3x| - 4 = 2x - 4 is to isolate the absolute value term. This means we need to get |3x| by itself on one side of the equation. To achieve this, we perform the opposite operation of what is being done to the absolute value term. In this case, we have “- 4” on the left side of the equation along with |3x|. The opposite operation of subtracting 4 is adding 4. Therefore, we add 4 to both sides of the equation to maintain the equality. This gives us: |3x| - 4 + 4 = 2x - 4 + 4. Simplifying both sides, we get |3x| = 2x. This step is crucial because it simplifies the equation and allows us to focus solely on the absolute value expression. By isolating |3x|, we can now clearly see the two possibilities we need to consider: 3x can be either equal to 2x or equal to -2x, depending on whether 3x is positive or negative. This isolation process is a fundamental technique in solving absolute value equations and lays the groundwork for the subsequent steps. Without isolating the absolute value term, it would be significantly more challenging to address the two possible cases and arrive at the correct solutions. This foundational step sets the stage for a systematic approach to solving the equation.

Step 2: Split the Equation into Two Cases

Now that we have isolated the absolute value term, |3x| = 2x, we must consider the two possible scenarios that arise due to the nature of the absolute value function. The absolute value of a quantity is its distance from zero, which means that the quantity inside the absolute value can be either positive or negative. This gives rise to two distinct cases that we must address separately to find all possible solutions. Case 1: 3x ≥ 0. In this case, the expression inside the absolute value is non-negative, so |3x| is simply equal to 3x. Therefore, our equation becomes 3x = 2x. This is a linear equation that can be easily solved by subtracting 2x from both sides, yielding x = 0. This value is a potential solution, but it must be checked in the original equation to ensure it is not extraneous. Case 2: 3x < 0. In this case, the expression inside the absolute value is negative, so |3x| is equal to the negation of 3x, which is -3x. Therefore, our equation becomes -3x = 2x. This is another linear equation that can be solved by adding 3x to both sides, resulting in 0 = 5x. Dividing both sides by 5, we get x = 0. However, we must also consider the condition 3x < 0, which means x < 0. Since x = 0 does not satisfy x < 0, we need to solve the equation -3x = 2x under the assumption that x < 0. Adding 3x to both sides gives 0 = 5x, and dividing by 5 yields x = 0. But this contradicts our assumption that x < 0, so we need to consider the implications of this contradiction carefully. Alternatively, we can analyze the equation -3x = 2x directly. To solve this, we can add 3x to both sides, resulting in 0 = 5x. Dividing both sides by 5 gives us x = 0. Since this solution does not satisfy the condition 3x < 0 (because 3 * 0 = 0, which is not less than 0), it is not a valid solution for this case. This careful consideration of the conditions and the implications of each case is crucial for accurately solving absolute value equations.

Step 3: Solve Each Case Separately

Having split the absolute value equation |3x| - 4 = 2x - 4 into two cases, we now proceed to solve each case individually. This step involves applying algebraic techniques to isolate the variable x and determine the potential solutions for each scenario. Case 1: 3x ≥ 0, which implies |3x| = 3x. Substituting this into the equation |3x| = 2x, we get 3x = 2x. To solve this linear equation, we subtract 2x from both sides: 3x - 2x = 2x - 2x, which simplifies to x = 0. This is a potential solution, but we must verify that it satisfies the condition 3x ≥ 0. Since 3 * 0 = 0, and 0 is indeed greater than or equal to 0, x = 0 is a valid solution for this case. Case 2: 3x < 0, which implies |3x| = -3x. Substituting this into the equation |3x| = 2x, we get -3x = 2x. To solve this linear equation, we add 3x to both sides: -3x + 3x = 2x + 3x, which simplifies to 0 = 5x. Dividing both sides by 5 gives us x = 0. However, we must verify that this solution satisfies the condition 3x < 0. Since 3 * 0 = 0, and 0 is not less than 0, x = 0 does not satisfy the condition for this case. Therefore, there is no solution in this case that meets the initial requirement of 3x < 0. In summary, after solving both cases, we find that the only potential solution is x = 0 from Case 1. It is crucial to remember that each solution obtained from the separate cases must be checked against the original conditions and the original equation to ensure its validity. This meticulous approach ensures that we only accept solutions that are both mathematically sound and consistent with the initial problem.

Step 4: Check for Extraneous Solutions

After solving the two cases of the absolute value equation |3x| - 4 = 2x - 4, we have identified a potential solution: x = 0. However, it is crucial to verify this solution in the original equation to ensure it is not an extraneous solution. Extraneous solutions can arise when dealing with absolute value equations because the process of squaring or splitting the equation into cases can sometimes introduce solutions that do not satisfy the original equation. To check for extraneous solutions, we substitute the potential solution (x = 0) back into the original equation: |3x| - 4 = 2x - 4. Substituting x = 0, we get |3(0)| - 4 = 2(0) - 4. This simplifies to |0| - 4 = 0 - 4, which further simplifies to 0 - 4 = -4. Thus, we have -4 = -4, which is a true statement. This confirms that x = 0 is indeed a valid solution to the original equation and is not an extraneous solution. Since we found only one potential solution and it has been verified, we can confidently conclude that x = 0 is the solution to the absolute value equation |3x| - 4 = 2x - 4. The process of checking for extraneous solutions is a critical step in solving absolute value equations, as it ensures the accuracy and validity of the final answer. By rigorously verifying each potential solution, we can avoid the pitfall of including solutions that do not satisfy the original problem, providing a complete and reliable solution.

Final Answer: Verifying the Solution

Having gone through the steps of isolating the absolute value term, splitting the equation into cases, solving each case separately, and checking for extraneous solutions, we have arrived at the solution x = 0 for the absolute value equation |3x| - 4 = 2x - 4. To provide a comprehensive and conclusive answer, it is beneficial to reiterate the solution and confirm its validity once more. The solution we found is x = 0. Let's substitute this value back into the original equation to ensure it holds true: |3(0)| - 4 = 2(0) - 4. Simplifying the left side, we get |0| - 4 = 0 - 4 = -4. Simplifying the right side, we get 2(0) - 4 = 0 - 4 = -4. Since both sides of the equation are equal (-4 = -4), the solution x = 0 is verified as correct. Therefore, the final answer to the equation |3x| - 4 = 2x - 4 is x = 0. This thorough verification process underscores the importance of accuracy and attention to detail in solving mathematical problems, particularly those involving absolute values. By consistently checking our solutions, we can ensure that our answers are not only mathematically sound but also practically applicable. This comprehensive approach instills confidence in the solution and provides a clear and definitive answer to the problem.

Conclusion: Summarizing the Solution Process

In conclusion, solving the absolute value equation |3x| - 4 = 2x - 4 requires a systematic approach that involves several key steps. First, we isolated the absolute value term by adding 4 to both sides of the equation, resulting in |3x| = 2x. Then, we split the equation into two cases to address the two possible scenarios arising from the absolute value: 3x ≥ 0 and 3x < 0. For Case 1, where 3x ≥ 0, we substituted |3x| with 3x and solved the resulting linear equation 3x = 2x, which gave us x = 0. We verified that this solution satisfies the condition 3x ≥ 0. For Case 2, where 3x < 0, we substituted |3x| with -3x and solved the resulting linear equation -3x = 2x. This also gave us x = 0, but this solution did not satisfy the condition 3x < 0, so it was not a valid solution for this case. Finally, we checked the potential solution x = 0 in the original equation |3x| - 4 = 2x - 4 and confirmed that it is indeed a valid solution. Therefore, the solution to the absolute value equation |3x| - 4 = 2x - 4 is x = 0. This step-by-step process highlights the importance of understanding the properties of absolute values, splitting equations into appropriate cases, solving linear equations, and verifying solutions to avoid extraneous results. By following these steps, we can confidently solve a wide range of absolute value equations. The ability to solve such equations is crucial in various mathematical and real-world applications, making this a valuable skill for students and professionals alike.