Solving Absolute Value Equations A Step-by-Step Guide To 2 - 8|2x + 1| = -14

Hey guys! Ever stumbled upon an equation that looks a bit intimidating at first glance? Well, today we're going to tackle one such equation together: 2 - 8|2x + 1| = -14. Don't worry; we'll break it down step-by-step, so it feels like a breeze. We will delve into the solutions for this equation, providing a clear and detailed explanation to help you understand each step involved.

Understanding Absolute Value Equations

Before we dive into the solution, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero, regardless of whether the number is positive or negative. So, |5| is 5, and |-5| is also 5. This concept is crucial when dealing with absolute value equations because it means the expression inside the absolute value bars can be either positive or negative, leading to two possible scenarios that we need to consider. In tackling absolute value equations, it's essential to remember that the absolute value expression represents a distance, which is always non-negative. This understanding forms the basis for solving equations involving absolute values. When we encounter an equation like |2x + 1| = a, we need to consider two separate cases: 2x + 1 = a and 2x + 1 = -a. This is because both 'a' and '-a' have the same distance from zero. Therefore, solving absolute value equations involves carefully considering both positive and negative possibilities for the expression inside the absolute value bars. This approach ensures that we capture all possible solutions and avoid missing any potential answers. By understanding this fundamental principle, we can confidently navigate through the steps required to solve complex equations involving absolute values. So, always remember to split the problem into two cases based on the positive and negative values of the expression within the absolute value, and you'll be well on your way to solving any absolute value equation that comes your way.

Step-by-Step Solution

Alright, let's get our hands dirty and solve the equation 2 - 8|2x + 1| = -14. Our main goal here is to isolate the absolute value term, which in this case is |2x + 1|. Think of it like peeling an onion – we need to get rid of the outer layers first. To kick things off, we'll subtract 2 from both sides of the equation. This keeps the equation balanced and helps us move closer to isolating the absolute value term. Doing so gives us -8|2x + 1| = -16. See? We're already making progress! The next step is to divide both sides by -8. This will completely isolate the absolute value term on one side of the equation. When we perform this division, we get |2x + 1| = 2. Now, the equation looks much simpler, doesn't it? This is where the magic of absolute values comes into play. Remember how we talked about absolute value representing distance from zero? This means that the expression inside the absolute value bars, 2x + 1, can be either 2 or -2 because both of these numbers are a distance of 2 away from zero. So, we've transformed our original equation into two separate, more manageable equations: 2x + 1 = 2 and 2x + 1 = -2. These two equations represent the two possible scenarios that satisfy the original absolute value equation. By addressing both possibilities, we ensure that we capture all potential solutions. In the following steps, we will solve each of these equations independently, which will lead us to the final solutions for x. By breaking down the problem in this way, we make it much easier to understand and solve, even if you're new to absolute value equations. So, let's keep going and see what values of x we can find!

Case 1: 2x + 1 = 2

Let's dive into our first case: 2x + 1 = 2. This equation is now a simple linear equation, which we can solve using basic algebraic techniques. Our aim here is to isolate x on one side of the equation to find its value. The first step in this process is to subtract 1 from both sides of the equation. This maintains the balance of the equation while helping us move closer to isolating the term with x. Subtracting 1 from both sides gives us 2x = 1. We're almost there! Now, to completely isolate x, we need to get rid of the 2 that's multiplying it. We can do this by dividing both sides of the equation by 2. This step ensures that we are left with x alone on one side, revealing its value. Performing this division, we find that x = 1/2. So, we have our first potential solution! But remember, we're dealing with an absolute value equation, which means we need to consider another case as well. This solution, x = 1/2, represents one of the values that satisfies the original equation. However, to ensure we haven't missed any solutions, we need to address the other possibility that arises from the absolute value. This is why we split the problem into two cases in the first place. By systematically working through each case, we can be confident that we have found all possible solutions to the equation. So, let's keep this solution in mind and move on to the second case, where we'll explore the other potential value of x that satisfies the original equation. By considering both cases, we ensure a thorough and accurate solution to the problem.

Case 2: 2x + 1 = -2

Now, let's tackle our second case: 2x + 1 = -2. Just like in the first case, we have a linear equation to solve, and our goal remains the same – to isolate x. We'll follow a similar process as before, using algebraic manipulations to get x by itself on one side of the equation. To begin, we subtract 1 from both sides of the equation. This step helps us to isolate the term containing x and keeps the equation balanced. When we subtract 1 from both sides, we get 2x = -3. We're getting closer to our solution! Now, to completely isolate x, we need to get rid of the 2 that's multiplying it. To do this, we divide both sides of the equation by 2. This step will leave us with x on one side and its value on the other. Performing the division, we find that x = -3/2. So, we have our second potential solution! This value of x, along with the one we found in Case 1, represents the complete set of solutions for the original equation. By carefully considering both positive and negative possibilities arising from the absolute value, we have ensured that we haven't missed any solutions. This methodical approach is key to solving absolute value equations accurately. Now that we have found both potential solutions, it's a good practice to check them in the original equation to make sure they are valid. This step can help us catch any errors that might have occurred during the solving process. In the next section, we'll summarize our findings and present the complete solution set for the equation. By breaking down the problem into manageable cases and systematically solving each one, we have successfully navigated this absolute value equation.

Checking the Solutions

To ensure we haven't made any mistakes, it's always a good idea to check our solutions in the original equation. Let's start with x = 1/2. Plugging this value into the equation 2 - 8|2x + 1| = -14, we get 2 - 8|2(1/2) + 1| = 2 - 8|1 + 1| = 2 - 8|2| = 2 - 16 = -14. Bingo! It checks out. This confirms that x = 1/2 is indeed a valid solution to the equation. Now, let's check our second solution, x = -3/2. Plugging this value into the equation, we get 2 - 8|2(-3/2) + 1| = 2 - 8|-3 + 1| = 2 - 8|-2| = 2 - 8(2) = 2 - 16 = -14. Great! This one checks out too. By verifying both solutions in the original equation, we can be confident that we have found the correct answers. This step is particularly important when dealing with absolute value equations, as it helps to identify any extraneous solutions that might have arisen due to the nature of the absolute value. In this case, both of our solutions are valid, but it's not always the case. Sometimes, plugging a potential solution back into the original equation will reveal that it doesn't actually satisfy the equation, and such a solution would need to be discarded. So, always remember to check your solutions, especially when working with absolute value equations. This simple step can save you from making mistakes and ensure that you arrive at the correct solution set. Now that we've checked our solutions and confirmed their validity, we can confidently present our final answer.

Final Answer

Alright, we've reached the finish line! After carefully working through the steps and verifying our answers, we can confidently say that the solutions to the equation 2 - 8|2x + 1| = -14 are x = 1/2 and x = -3/2. These are the two values of x that satisfy the given equation. Remember, when dealing with absolute value equations, it's crucial to consider both the positive and negative cases of the expression inside the absolute value bars. This approach ensures that you capture all possible solutions. By breaking down the problem into manageable steps, we were able to isolate the absolute value term, create two separate equations, and solve each one individually. This methodical approach is key to successfully solving these types of equations. And, of course, we didn't forget to check our solutions to make sure they were valid! Checking solutions is a crucial step in any equation-solving process, as it helps to catch any potential errors. So, there you have it! We've successfully solved the equation and found both solutions. I hope this step-by-step guide has been helpful in understanding how to tackle absolute value equations. Keep practicing, and you'll become a pro in no time! Remember, math can be fun when you break it down and approach it with a clear strategy. Keep up the great work, guys!

The solutions are: x = 1/2, -3/2