Solving Absolute Value Equations 4|x-3|-8=8 A Step-by-Step Guide

Navigating the world of absolute value equations can seem daunting, but with a systematic approach, even the most complex problems become manageable. In this comprehensive guide, we will dissect the equation 4|x-3|-8=8, providing a clear and concise pathway to the solution. We'll explore the fundamental principles of absolute value, demonstrate each step in detail, and equip you with the skills to confidently tackle similar equations in the future. Understanding how to solve absolute value equations is a crucial skill in algebra and beyond, laying the groundwork for more advanced mathematical concepts. Let's embark on this journey together, transforming a potentially challenging problem into an opportunity for mastery. Before diving into the specific steps, let's first solidify our understanding of absolute value itself. The absolute value of a number represents its distance from zero on the number line, regardless of direction. This means that the absolute value of a positive number is the number itself, and the absolute value of a negative number is its positive counterpart. For example, |5| = 5 and |-5| = 5. This concept is critical for solving equations involving absolute values because it introduces the possibility of two distinct cases: the expression inside the absolute value bars can be either positive or negative. To effectively solve 4|x-3|-8=8, we must consider both scenarios, ensuring we capture all possible solutions. Our approach will involve isolating the absolute value expression, setting up two separate equations based on the two cases, and solving each equation individually. By meticulously following this process, we can confidently arrive at the correct solutions and gain a deeper understanding of absolute value equations. Now, let's begin the step-by-step solution, starting with isolating the absolute value term.

Step 1: Isolating the Absolute Value

The initial step in solving the equation 4|x-3|-8=8 involves isolating the absolute value term. This is crucial because it allows us to directly address the two possible cases arising from the absolute value function. To isolate the term, we need to eliminate any constants or coefficients that are outside the absolute value bars. In this case, we have a constant term, -8, and a coefficient, 4, multiplying the absolute value expression. We'll start by addressing the constant term. To eliminate -8, we perform the inverse operation, which is adding 8 to both sides of the equation. This maintains the equality and moves us closer to isolating the absolute value. Adding 8 to both sides of the equation 4|x-3|-8=8 gives us 4|x-3| = 16. Now, the constant term is eliminated from the left side, leaving us with only the term containing the absolute value. The next step is to address the coefficient, 4, that is multiplying the absolute value expression. To eliminate this coefficient, we again perform the inverse operation, which is division. We divide both sides of the equation by 4, maintaining the equality and isolating the absolute value completely. Dividing both sides of 4|x-3| = 16 by 4 results in |x-3| = 4. At this point, we have successfully isolated the absolute value expression. This is a significant milestone because it allows us to proceed to the next stage, which involves considering the two cases that arise from the absolute value. Remember, the absolute value of an expression can be either positive or negative, so we must account for both possibilities to find all solutions. With the absolute value isolated, we are now ready to set up the two cases and solve for x. This methodical approach ensures that we don't miss any potential solutions and that we understand the underlying principles of absolute value equations. Let's move on to the next step, where we'll explore the two cases in detail.

Step 2: Setting Up Two Cases

Having successfully isolated the absolute value expression in the equation |x-3| = 4, we now arrive at a pivotal point: setting up the two cases. This step is fundamental to solving absolute value equations because it acknowledges the inherent duality of the absolute value function. The absolute value of a number represents its distance from zero, meaning that the expression inside the absolute value bars can be either positive or negative while still resulting in the same absolute value. In our case, the equation |x-3| = 4 tells us that the expression x-3 is 4 units away from zero. This can happen in two ways: either x-3 is equal to 4, or x-3 is equal to -4. These two possibilities form the basis of our two cases. Case 1: x-3 = 4. This case considers the scenario where the expression inside the absolute value bars is positive. It directly equates the expression x-3 to the positive value 4. This equation represents one possible relationship between x and 3 that satisfies the original absolute value equation. Case 2: x-3 = -4. This case addresses the scenario where the expression inside the absolute value bars is negative. It equates the expression x-3 to the negative value -4. This equation represents the other possible relationship between x and 3 that satisfies the original absolute value equation. By setting up these two cases, we create two separate linear equations that we can solve independently. Each equation will yield a potential solution for x. It is crucial to consider both cases to ensure that we find all possible solutions to the original absolute value equation. Failing to account for either case would result in an incomplete solution. Now that we have established the two cases, the next step is to solve each equation individually. This will involve applying basic algebraic principles to isolate x in each case. Let's proceed to the next section, where we will delve into the solution process for each case.

Step 3: Solving Case 1 (x-3 = 4)

Having established Case 1 as x-3 = 4, our next task is to solve this linear equation for x. Solving for x involves isolating the variable on one side of the equation. In this case, x is currently being subtracted by 3. To isolate x, we need to perform the inverse operation, which is addition. We will add 3 to both sides of the equation, maintaining the equality and effectively canceling out the -3 on the left side. Adding 3 to both sides of x-3 = 4 gives us x-3+3 = 4+3. Simplifying both sides, we get x = 7. This is the solution for x in Case 1. It indicates that when x is equal to 7, the expression x-3 equals 4, which satisfies the original absolute value equation |x-3| = 4. To verify this solution, we can substitute x = 7 back into the original equation and check if it holds true. Substituting x = 7 into |x-3| = 4 gives us |7-3| = 4, which simplifies to |4| = 4. Since the absolute value of 4 is indeed 4, our solution x = 7 is verified as correct for Case 1. This process of verification is crucial in mathematics, as it ensures that the solutions we obtain are valid and accurate. It provides confidence in our work and helps prevent errors. Now that we have successfully solved Case 1, we have one potential solution for the original absolute value equation. However, we must remember that there is also a Case 2 to consider. Case 2 represents the scenario where the expression inside the absolute value bars is negative. To obtain a complete solution to the original equation, we must solve Case 2 as well. Let's move on to the next section, where we will tackle Case 2 and determine the other potential solution for x.

Step 4: Solving Case 2 (x-3 = -4)

Having solved Case 1 and obtained the solution x = 7, we now turn our attention to Case 2, which is represented by the equation x-3 = -4. This case considers the scenario where the expression x-3 inside the absolute value bars is negative. To solve for x in this equation, we employ the same algebraic principle as in Case 1: isolating the variable by performing the inverse operation. In this instance, x is being subtracted by 3, so we add 3 to both sides of the equation to eliminate the -3 on the left side and maintain the equality. Adding 3 to both sides of x-3 = -4 gives us x-3+3 = -4+3. Simplifying both sides, we arrive at x = -1. This is the solution for x in Case 2. It indicates that when x is equal to -1, the expression x-3 equals -4, which also satisfies the original absolute value equation |x-3| = 4 because the absolute value of -4 is 4. Just as we did in Case 1, we should verify this solution to ensure its accuracy. Substituting x = -1 back into the original equation |x-3| = 4 gives us |-1-3| = 4, which simplifies to |-4| = 4. Since the absolute value of -4 is indeed 4, our solution x = -1 is verified as correct for Case 2. This confirms that we have found a valid solution for the second possible scenario arising from the absolute value equation. With both Case 1 and Case 2 solved, we have now identified two potential solutions for the original equation 4|x-3|-8=8: x = 7 from Case 1 and x = -1 from Case 2. The final step is to summarize these solutions and present them as the complete solution set for the equation. Let's move on to the next section, where we will consolidate our findings and state the final answer.

Step 5: Combining the Solutions and Final Answer

After meticulously solving both Case 1 and Case 2 of the equation 4|x-3|-8=8, we have arrived at two potential solutions: x = 7 and x = -1. These solutions represent the values of x that satisfy the original equation, considering both the positive and negative possibilities arising from the absolute value. To ensure the completeness and accuracy of our solution, it is essential to combine these individual solutions into a single solution set. This set represents all possible values of x that make the equation true. In this case, our solution set consists of the two values we found: 7 and -1. Therefore, the complete solution to the equation 4|x-3|-8=8 is x = 7 or x = -1. We can express this solution in various ways, such as using set notation {-1, 7} or simply stating that the solutions are x = -1 and x = 7. It is important to present the solutions clearly and concisely, making it easy for others to understand the answer. By following this step-by-step process, we have not only solved the equation but also gained a deeper understanding of how absolute value equations work. We have learned to isolate the absolute value term, set up and solve two cases, and combine the solutions into a comprehensive answer. This approach can be applied to a wide range of absolute value equations, empowering us to tackle more complex problems with confidence. In conclusion, the solutions to the equation 4|x-3|-8=8 are x = -1 and x = 7. This completes our journey through the solution process, providing a clear and detailed explanation of each step. We have successfully unlocked the solution to this absolute value equation, demonstrating the power of systematic problem-solving and a thorough understanding of mathematical principles.