Solving Absolute Value Equations A Step By Step Guide

Hey guys! Today, we are going to dive into solving an absolute value equation. Absolute value equations might seem a bit intimidating at first, but don't worry! We'll break it down step-by-step to make it super easy to understand. Our mission today is to solve for x in the equation 2|4-(5/2)x|+6=18. We'll go through the entire process, ensuring you grasp every concept along the way. This particular problem is a fantastic example of how to tackle absolute value equations, combining algebraic manipulation with an understanding of what absolute value truly means. So, grab your thinking caps, and let’s get started! Whether you're a student brushing up on your algebra skills or just someone who enjoys a good math challenge, this guide is here to help. We'll begin by isolating the absolute value term, then consider both positive and negative cases, and finally, we'll wrap up by verifying our solutions to ensure they're spot-on. By the end of this article, you'll not only know how to solve this specific equation but also have a solid foundation for handling similar problems in the future. Remember, math is like building blocks, each concept building upon the previous one. So, let’s lay a strong foundation today and conquer this equation together!

Step 1: Isolate the Absolute Value Term

Okay, so the first thing we need to do when solving an absolute value equation is to isolate the absolute value term. Think of it like clearing a path so we can get to the core of the problem. In our equation, 2|4-(5/2)x|+6=18, the absolute value term is |4-(5/2)x|. To isolate it, we need to get rid of everything else around it. The first thing we'll tackle is the +6 on the left side of the equation. To do this, we'll subtract 6 from both sides. Remember, whatever we do to one side of the equation, we have to do to the other side to keep things balanced! This is a fundamental principle in algebra, ensuring that the equality remains intact throughout our solving process. By applying this principle consistently, we can systematically simplify the equation and move closer to our solution. So, subtracting 6 from both sides, we get: 2|4-(5/2)x|+6-6=18-6. Simplifying this gives us 2|4-(5/2)x|=12. Great! We've made progress. Now, we still need to get the absolute value term completely by itself. Notice that the absolute value is being multiplied by 2. To undo this multiplication, we'll divide both sides of the equation by 2. Again, it's all about maintaining balance and performing the same operation on both sides. Dividing both sides by 2, we have (2|4-(5/2)x|)/2=12/2. This simplifies to |4-(5/2)x|=6. Awesome! We've successfully isolated the absolute value term. Now we're ready for the next big step: dealing with the absolute value itself. Isolating the absolute value is a crucial step because it sets us up to consider both possible scenarios: the expression inside the absolute value could be positive or negative, but its absolute value is the same. This is where the magic happens, and we start to see how absolute value equations have a unique twist compared to regular equations. So, with the absolute value isolated, we're perfectly positioned to move forward and uncover the solutions.

Step 2: Consider Both Positive and Negative Cases

Alright, now that we've isolated the absolute value term, it's time to tackle the heart of the problem. Remember, the absolute value of a number is its distance from zero, which means it can be either positive or negative inside the absolute value bars, but the result will always be positive. So, when we have an equation like |4-(5/2)x|=6, it means that the expression inside the absolute value, which is 4-(5/2)x, could be either 6 or -6. This is the key insight that allows us to solve absolute value equations. We need to consider both possibilities to find all possible solutions for x. Let's start with the positive case. If 4-(5/2)x is equal to 6, we can write the equation as 4-(5/2)x=6. This is a linear equation that we can solve using standard algebraic techniques. We'll subtract 4 from both sides to get -(5/2)x=2. Then, to isolate x, we'll multiply both sides by -2/5, which is the reciprocal of -5/2. This gives us x=2*(-2/5), which simplifies to x=-4/5. So, we have one potential solution: x=-4/5. But we're not done yet! We still need to consider the negative case. If 4-(5/2)x is equal to -6, we can write the equation as 4-(5/2)x=-6. Again, this is a linear equation that we can solve. We'll subtract 4 from both sides to get -(5/2)x=-10. Then, we'll multiply both sides by -2/5 to isolate x. This gives us x=-10*(-2/5), which simplifies to x=4. So, we have another potential solution: x=4. Now, we have two possible solutions for x: -4/5 and 4. But before we declare victory, there's one more crucial step we need to take: verifying our solutions. This is super important because sometimes we can get extraneous solutions, which are solutions that don't actually work in the original equation. So, let's move on to the verification step to make sure our solutions are the real deal.

Step 3: Verify the Solutions

Alright, we've arrived at a super important stage: verifying our solutions. This step is like the quality control of our math work, making sure that the values we found for x actually satisfy the original equation. It's easy to make small mistakes along the way, and plugging our solutions back into the original equation helps us catch any errors and avoid claiming a wrong answer. Plus, with absolute value equations, it's particularly important to check for extraneous solutions, which are values that pop up during the solving process but don't actually work when you plug them back into the original equation. So, let's dive into verifying our solutions. We found two potential solutions for x: -4/5 and 4. Our original equation is 2|4-(5/2)x|+6=18. We'll start by plugging in x=-4/5 into the equation. Substituting -4/5 for x, we get 2|4-(5/2)(-4/5)|+6. Let's simplify inside the absolute value: 4-(5/2)(-4/5) = 4 + 2 = 6. So, we have 2|6|+6. The absolute value of 6 is just 6, so we have 2*6+6 = 12+6 = 18. Yay! It checks out! So, x=-4/5 is definitely a solution. Now, let's check our other solution, x=4. Substituting 4 for x in the original equation, we get 2|4-(5/2)*4|+6. Let's simplify inside the absolute value: 4-(5/2)4 = 4-10 = -6. So, we have 2|-6|+6. The absolute value of -6 is 6, so we have 26+6 = 12+6 = 18. Awesome! This one checks out too! Both x=-4/5 and x=4 satisfy the original equation. That means we've found the correct solutions. Verifying solutions might seem like an extra step, but it's a crucial part of the problem-solving process. It gives us confidence in our answers and ensures that we're not led astray by extraneous solutions. So, always take the time to verify your solutions, especially in absolute value equations. With our solutions verified, we can confidently move on to the final step: stating our answer.

Step 4: State the Solution

Okay, we've done all the hard work! We've isolated the absolute value term, considered both positive and negative cases, and verified our solutions. Now comes the satisfying part: stating our final answer. It's like reaching the summit after a challenging climb – a moment to celebrate our accomplishment! So, let's recap what we've found. We were given the equation 2|4-(5/2)x|+6=18, and our mission was to solve for x. Through our step-by-step process, we discovered two solutions that satisfy this equation. These solutions are x=-4/5 and x=4. We meticulously verified each solution by plugging it back into the original equation, and we confirmed that both values make the equation true. This gives us the confidence to declare our final answer with certainty. So, drumroll please... The solutions for x in the equation 2|4-(5/2)x|+6=18 are x=-4/5 and x=4. We can write this concisely as x = -4/5, 4. And there you have it! We've successfully solved the absolute value equation. Stating the solution clearly is the final touch that completes the problem. It's like putting the period at the end of a sentence, signaling that our thought is complete. When presenting your answer, it's always a good idea to state it in a clear and organized manner, so there's no ambiguity. Whether you're writing it on a test, sharing it with a classmate, or explaining it to someone else, a clear statement of the solution makes your work shine. So, congratulations on making it to the end! You've mastered the art of solving this absolute value equation, and you've gained valuable skills that you can apply to other math problems. Now, go forth and conquer more equations!

Woo-hoo! We did it, guys! We successfully solved for x in the equation 2|4-(5/2)x|+6=18. You've now seen how to break down an absolute value equation into manageable steps, and you've learned the importance of isolating the absolute value term, considering both positive and negative cases, and verifying your solutions. These are crucial skills that will help you tackle a wide range of algebraic problems. Remember, practice makes perfect! The more you work with these types of equations, the more comfortable and confident you'll become. Don't be afraid to try different problems and challenge yourself. Math is a journey, and every problem you solve is a step forward. Keep practicing, keep exploring, and keep having fun with it! And remember, if you ever get stuck, there are tons of resources available to help you. Whether it's textbooks, online tutorials, or your friendly neighborhood math whiz, there's always someone or something that can guide you. So, embrace the challenge, and keep on solving! We started with a potentially intimidating equation, but we systematically broke it down and conquered it together. You've not only learned how to solve this specific problem but also gained a deeper understanding of absolute value equations in general. This kind of conceptual understanding is what truly empowers you to excel in math. So, take pride in your accomplishment, and carry this newfound knowledge forward. You've got this! Solving equations is like unlocking secrets, and with each equation you solve, you're unlocking a new level of mathematical understanding. So, keep those keys turning, and keep exploring the amazing world of math!