Solving Absolute Value Equations A Step-by-Step Guide To Solve |2.5x - 6.8| = 12.9
In this comprehensive guide, we will delve into the process of solving for x in the absolute value equation |2.5x - 6.8| = 12.9. Absolute value equations often present a unique challenge due to the nature of absolute values, which represent the distance of a number from zero. This means that the expression inside the absolute value can be either positive or negative, leading to two possible scenarios that we need to consider. Understanding how to handle these scenarios is crucial for accurately solving the equation. This article aims to provide a step-by-step solution, ensuring clarity and ease of understanding for readers of all mathematical backgrounds. We will break down the problem into manageable parts, explaining each step in detail, and provide insights into the underlying principles of absolute value equations. By the end of this guide, you will not only be able to solve this specific equation but also gain a solid foundation for tackling similar problems in the future. Whether you're a student looking to improve your algebra skills or simply someone interested in mathematical problem-solving, this article is designed to equip you with the knowledge and confidence to approach absolute value equations effectively.
Before we dive into the solution, it's essential to grasp the fundamental concept of absolute value. The absolute value of a number is its distance from zero on the number line, regardless of direction. This is why the absolute value is always non-negative. For instance, |5| = 5 and |-5| = 5. This principle is the cornerstone of solving absolute value equations. When we encounter an equation like |2.5x - 6.8| = 12.9, it implies that the expression inside the absolute value, which is 2.5x - 6.8, can be either 12.9 or -12.9, because both values have an absolute value of 12.9. This duality is what necessitates splitting the original equation into two separate equations, each representing one of these possibilities. Ignoring this crucial step can lead to an incomplete or incorrect solution. Therefore, a clear understanding of this concept is paramount for effectively tackling absolute value equations. This section will further explore the implications of absolute value in the context of equations and how it shapes our approach to solving them. By mastering this foundational knowledge, you will be well-prepared to address the specific equation at hand and similar problems in the future. The ability to recognize and apply the properties of absolute value is a valuable skill in algebra and beyond.
The first critical step in solving the absolute value equation |2.5x - 6.8| = 12.9 is to recognize the dual nature of absolute values and split the equation into two distinct cases. This is because the expression inside the absolute value bars, 2.5x - 6.8, can be either equal to 12.9 or -12.9, as both of these values have an absolute value of 12.9. Case 1 represents the scenario where the expression is equal to the positive value, 12.9, resulting in the equation 2.5x - 6.8 = 12.9. This case addresses the situation where the expression inside the absolute value is already positive or zero. Case 2, on the other hand, considers the possibility where the expression is equal to the negative value, -12.9, leading to the equation 2.5x - 6.8 = -12.9. This case is crucial because it accounts for the scenario where the expression inside the absolute value is negative, and taking the absolute value makes it positive. By splitting the equation into these two cases, we ensure that we consider all possible solutions for x. This step is fundamental to solving absolute value equations accurately and completely. Each case will be solved independently in the subsequent steps, and the solutions obtained from both cases will form the complete solution set for the original equation. The ability to correctly split an absolute value equation is a key skill in algebra.
Now that we've established the two cases, let's focus on solving for x in Case 1: 2.5x - 6.8 = 12.9. The goal here is to isolate x on one side of the equation. We'll start by addressing the constant term, -6.8, on the left side. To do this, we add 6.8 to both sides of the equation. This maintains the equality and moves us closer to isolating x. The equation then becomes 2.5x = 12.9 + 6.8, which simplifies to 2.5x = 19.7. Next, to completely isolate x, we need to deal with the coefficient 2.5 that is multiplying x. We achieve this by dividing both sides of the equation by 2.5. This step is crucial as it directly solves for x. The equation now reads x = 19.7 / 2.5. Performing this division yields x = 7.88. Therefore, the solution for x in Case 1 is 7.88. This value represents one possible solution to the original absolute value equation. It's important to note that we're only halfway through the problem at this point, as we still need to solve for x in Case 2. However, by carefully following these algebraic steps, we've successfully found one solution. The process of isolating x through addition, subtraction, multiplication, and division is a fundamental skill in algebra, and this example demonstrates its application in the context of absolute value equations.
Having solved for x in Case 1, we now turn our attention to Case 2: 2.5x - 6.8 = -12.9. Similar to Case 1, our objective is to isolate x on one side of the equation. The first step in this process is to eliminate the constant term, -6.8, from the left side. We accomplish this by adding 6.8 to both sides of the equation, ensuring that the equality is maintained. This transforms the equation into 2.5x = -12.9 + 6.8. Simplifying the right side of the equation gives us 2.5x = -6.1. Now, to completely isolate x, we must remove the coefficient 2.5 that is multiplying x. We do this by dividing both sides of the equation by 2.5. This critical step directly solves for x and results in the equation x = -6.1 / 2.5. Performing the division yields x = -2.44. Thus, the solution for x in Case 2 is -2.44. This value represents the second possible solution to the original absolute value equation. We have now found solutions for x in both cases, which means we have accounted for all possibilities arising from the absolute value. The systematic approach of isolating x, using inverse operations, is a fundamental technique in algebra. By applying this method consistently, we can confidently solve for unknowns in various types of equations. With both solutions in hand, we are ready to move on to the final step of verifying our answers.
After solving for x in both cases of the absolute value equation, it's crucial to verify the solutions to ensure their accuracy. This step is particularly important in absolute value equations because extraneous solutions can sometimes arise due to the nature of the absolute value function. To check our solutions, we substitute each value of x back into the original equation, |2.5x - 6.8| = 12.9, and see if the equation holds true. Let's start with the solution from Case 1, x = 7.88. Substituting this value into the equation gives us |2.5(7.88) - 6.8|. First, we calculate 2.5 * 7.88 = 19.7. Then, we subtract 6.8 from 19.7, which results in 12.9. The absolute value of 12.9 is indeed 12.9, so this solution checks out. Now, let's verify the solution from Case 2, x = -2.44. Substituting this value into the equation gives us |2.5(-2.44) - 6.8|. First, we calculate 2.5 * -2.44 = -6.1. Then, we subtract 6.8 from -6.1, which results in -12.9. The absolute value of -12.9 is 12.9, so this solution also checks out. Since both solutions satisfy the original equation, we can confidently conclude that they are correct. This verification process underscores the importance of checking solutions, especially in the context of absolute value equations, to avoid accepting extraneous solutions. By taking the time to verify, we ensure the integrity of our solution and demonstrate a thorough understanding of the problem-solving process.
In conclusion, we have successfully solved the absolute value equation |2.5x - 6.8| = 12.9 by systematically breaking it down into manageable steps. We began by understanding the fundamental concept of absolute value, which is the distance of a number from zero, and how this leads to two possible cases in an equation. Recognizing this duality is crucial for accurately solving absolute value equations. We then split the equation into two distinct cases: 2.5x - 6.8 = 12.9 and 2.5x - 6.8 = -12.9. This step ensured that we considered all possible scenarios arising from the absolute value. Next, we solved for x in each case independently. In Case 1, we found x = 7.88, and in Case 2, we found x = -2.44. The process of isolating x in each case involved applying inverse operations, a fundamental skill in algebra. Finally, we verified our solutions by substituting them back into the original equation. This crucial step confirmed that both x = 7.88 and x = -2.44 are valid solutions, and it also highlighted the importance of checking for extraneous solutions in absolute value equations. By following this step-by-step approach, we have not only solved the specific equation but also demonstrated a robust method for tackling similar problems in the future. The ability to confidently solve absolute value equations is a valuable asset in mathematics, and this guide has aimed to provide a clear and comprehensive understanding of the process.
