Solving Absolute Value Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of absolute value equations. Specifically, we're going to solve for v in the equation $|-8-5v| = |5v+7|$. This might seem a bit intimidating at first, but trust me, we'll break it down step by step. Understanding how to solve absolute value equations is a crucial skill in algebra, and it's super important for anyone looking to master math. This guide will walk you through the process, making sure you understand every single detail. Let's get started! So, what does it mean to solve an absolute value equation? Simply put, it means finding all the values of the variable (in this case, v) that make the equation true. The absolute value of a number is its distance from zero on the number line, and this distance is always non-negative. This means that both positive and negative numbers can have the same absolute value. For example, the absolute value of 3 is 3, and the absolute value of -3 is also 3. That's why when we solve these equations, we usually end up with two different possibilities to consider. Let's get into the meat of the problem, focusing on how we can tackle absolute value equations. The equation we are trying to solve is $|-8 - 5v| = |5v + 7|$.

Understanding Absolute Value Equations

Alright, before we jump into the calculations, let's get our heads around what we're actually dealing with. Absolute value equations are those bad boys that involve the absolute value of an expression. The absolute value of a number is its distance from zero, no matter the direction. This is always non-negative, which means we have to take two different paths to solve these types of equations. The absolute value bars essentially force the expression inside to be positive. This is the key concept to understand before we start solving. Remember, the absolute value of something is its distance from zero. Now, because distance can be in either direction (positive or negative), we always have to consider two scenarios when solving equations like the one we're looking at. For our particular equation, $|-8 - 5v| = |5v + 7|$, there are two main cases to consider. Think of it like a fork in the road! To correctly solve absolute value equations like these, we need to handle two separate cases. The first case is when the expressions inside the absolute value symbols are equal to each other, i.e., $-8 - 5v = 5v + 7$. The second case is when one expression is equal to the negative of the other, i.e., $-8 - 5v = -(5v + 7)$. We must solve each of these to make sure we find all possible solutions. Ignoring these scenarios will lead to incorrect answers. It is important to verify each solution in the original equation to catch any extraneous solutions. We'll go through this step-by-step to make sure we don't miss anything!

Case 1: The Expressions are Equal

Let's start with the first case: $-8 - 5v = 5v + 7$. This means the expressions inside the absolute value signs are equal. So, we simply remove the absolute value signs and solve for v. First, let's add 5v to both sides of the equation. This gives us: $-8 - 5v + 5v = 5v + 7 + 5v$. This simplifies to: $-8 = 10v + 7$. Next, let's subtract 7 from both sides: $-8 - 7 = 10v + 7 - 7$. This simplifies to: $-15 = 10v$. Now, to isolate v, we'll divide both sides by 10: $-15 / 10 = 10v / 10$. This gives us v = -1.5. We're not done yet, guys! It's super important to check this solution. We need to plug this value back into the original equation to make sure it works. So, let's plug v = -1.5 into $|-8 - 5v| = |5v + 7|$: $|-8 - 5(-1.5)| = |5(-1.5) + 7|$. Simplifying, we get: $|-8 + 7.5| = |-7.5 + 7|$, which turns into $|-0.5| = |-0.5|$. The absolute value of -0.5 is 0.5, so we get: 0.5 = 0.5. Since this is true, v = -1.5 is a valid solution. Nice! This is how you would proceed to solve the absolute value equations.

Case 2: One Expression is the Negative of the Other

Alright, let's now look at our second case. This happens when one of the expressions inside the absolute value is the negative of the other. This means we set up our equation like this: $-8 - 5v = -(5v + 7)$. Let's simplify this: $-8 - 5v = -5v - 7$. Now, let's add 5v to both sides: $-8 - 5v + 5v = -5v - 7 + 5v$. This simplifies to: $-8 = -7$. Wait a second... this isn't right! We've ended up with a statement that's mathematically impossible. $-8$ can never equal $-7$. This indicates that there is no solution from this case. So, the second case doesn't give us any valid solutions. That’s perfectly normal, and it happens sometimes in absolute value equations. Remember, the process is about covering all possible paths and finding the values of v that make the original equation true.

Putting It All Together and The Final Answer

Alright, we've done the work! We've solved our two cases and now we need to combine our findings. From our first case, we got v = -1.5 as a solution. The second case gave us no solutions. Therefore, the only solution to the equation $|-8 - 5v| = |5v + 7|$ is v = -1.5. So, to answer the original question, you'd just write -1.5. Always check your solutions in the original equation! This step is super important to ensure the validity of the solution. It is a critical step to make sure our calculations are accurate. It helps us catch any errors in our algebra. So, always make sure you substitute your solution back into the equation to verify it. Also, remember that the absolute value of a number is its distance from zero, and that's why it can never be negative. Understanding this basic principle is key to solving absolute value equations. Keep practicing, and you'll master these problems in no time! You've got this, and keep up the great work!