Solve |x-4|+6=15: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of absolute value equations. These might seem a bit tricky at first, but trust me, once you get the hang of them, they're actually pretty straightforward. We're going to tackle the equation $|x-4|+6=15$ step-by-step, so you'll not only get the answer but also understand the why behind each move. This equation is a classic example that helps illustrate the fundamental principles of solving absolute value problems. So, whether you're a student prepping for an exam or just someone looking to brush up on your math skills, this guide is for you. We’ll break down each step, making sure you understand the logic and reasoning behind it. This way, you’ll be able to confidently tackle similar problems in the future. Remember, math isn't about memorizing steps; it's about understanding concepts. And that's exactly what we're aiming for here. Let’s jump right into it and unravel the mystery of absolute value equations together! We'll cover everything from isolating the absolute value expression to handling the two possible scenarios that arise due to the nature of absolute values. So, grab a pen and paper, and let's get started!

Understanding Absolute Value

Before we jump into solving the equation, let's quickly recap what absolute value actually means. The absolute value of a number is its distance from zero on the number line. This means that it's always non-negative. For example, the absolute value of 5, written as $|5|$, is 5 because 5 is 5 units away from zero. Similarly, the absolute value of -5, written as $|-5|$, is also 5 because -5 is also 5 units away from zero. This concept is crucial for understanding how to solve absolute value equations. When we see an expression inside absolute value bars, we need to consider two possibilities: the expression inside could be positive or negative, but its distance from zero will always be positive. This dual nature is what makes solving absolute value equations a bit different from solving regular equations. We have to account for both scenarios to find all possible solutions. Think of it like this: absolute value strips away the sign, leaving only the magnitude. So, when we solve an absolute value equation, we’re essentially figuring out what values of the variable would make the expression inside the absolute value bars have a certain magnitude. This understanding forms the bedrock of our problem-solving approach. Without grasping this fundamental concept, the steps we take to solve the equation might seem arbitrary. But with this understanding, each step will make logical sense. So, keep this in mind as we proceed: absolute value is all about distance from zero, and distance is always non-negative. This is the key to unlocking absolute value equations.

Isolating the Absolute Value

Okay, now that we've refreshed our understanding of absolute value, let's get to the first step in solving our equation: $|x-4|+6=15$. The very first thing we need to do is to isolate the absolute value expression, which in this case is $|x-4|$. This means we want to get the absolute value expression all by itself on one side of the equation. To do this, we need to get rid of the '+6' that's hanging out on the left side. How do we do that? Simple! We use the inverse operation. Since we're adding 6, we'll subtract 6 from both sides of the equation. This is a fundamental principle in algebra: whatever you do to one side of the equation, you must do to the other to keep it balanced. Subtracting 6 from both sides gives us: $|x-4| + 6 - 6 = 15 - 6$. This simplifies to $|x-4| = 9$. Great! We've successfully isolated the absolute value expression. Now we're one step closer to solving for x. This isolation step is crucial because it sets us up to handle the two possibilities that arise from the absolute value. Once the absolute value expression is isolated, we can consider the two cases: the expression inside the absolute value bars is either equal to the positive value on the other side of the equation, or it's equal to the negative value. This is where the magic happens, and we start to see how absolute value equations reveal their solutions. So, remember, always start by isolating the absolute value expression. It's the key to unlocking the rest of the problem.

Setting Up Two Equations

Alright, guys, we've successfully isolated the absolute value, giving us $|x-4|=9$. Now comes the really cool part! Because of the nature of absolute value, there are actually two possibilities we need to consider. Remember, the absolute value of a number is its distance from zero. So, if $|x-4|$ equals 9, that means the expression $(x-4)$ is 9 units away from zero. This can happen in two ways: either $(x-4)$ is equal to 9, or $(x-4)$ is equal to -9. This is the crux of solving absolute value equations: we split the single equation into two separate equations to account for both positive and negative possibilities. So, let's write out our two equations:

1. $x - 4 = 9$
2. $x - 4 = -9$

See how we've taken the expression inside the absolute value bars and set it equal to both the positive and negative values of the number on the other side of the equation? This is the key step in handling absolute values. We've now transformed one tricky absolute value equation into two simple linear equations that we can easily solve. Each of these equations represents a possible scenario that satisfies the original absolute value equation. By solving both equations, we'll find all possible values of x that make the original equation true. This approach might seem a bit unusual at first, but it's a direct consequence of the definition of absolute value. Remember, absolute value is about distance from zero, and a certain distance can be achieved in two directions: positive and negative. So, setting up these two equations is not just a mathematical trick; it's a reflection of the underlying concept of absolute value. Now that we have our two equations, let's move on to solving them and finding our solutions.

Solving the Equations

Now that we have our two equations, let's solve them one by one. This part is pretty straightforward, just like solving any regular linear equation. Let's start with the first equation:

Equation 1: $x - 4 = 9$

To solve for x, we need to isolate it. We can do this by adding 4 to both sides of the equation: $x - 4 + 4 = 9 + 4$. This simplifies to $x = 13$. Awesome! We've found our first solution.

Now, let's move on to the second equation:

Equation 2: $x - 4 = -9$

Again, we need to isolate x. We do this by adding 4 to both sides of the equation: $x - 4 + 4 = -9 + 4$. This simplifies to $x = -5$. Fantastic! We've found our second solution.

So, we have two possible values for x: 13 and -5. These are the solutions to our absolute value equation. Solving these equations is a crucial step, as it's where we actually determine the numerical values of x that satisfy the original equation. Each equation represents a different scenario, and by solving both, we ensure that we've accounted for all possibilities. The process of solving each equation involves using basic algebraic principles, such as adding the same value to both sides to maintain equality. This is a fundamental technique in algebra, and it's essential for solving a wide range of equations. In this case, adding 4 to both sides allowed us to isolate x and reveal its value. So, remember, when solving absolute value equations, don't forget to split the equation into two cases and solve each one separately. This is the key to finding all the solutions.

Checking the Solutions

Before we declare victory and move on, it's super important to check our solutions. This is a crucial step in solving any equation, but it's especially important with absolute value equations. Why? Because sometimes, we might get solutions that don't actually work in the original equation, known as extraneous solutions. To check our solutions, we simply plug each value of x back into the original equation and see if it holds true.

Let's check $x = 13$:

Original equation: $|x - 4| + 6 = 15$

Substitute $x = 13$: $|13 - 4| + 6 = 15$

Simplify: $|9| + 6 = 15$

Further simplification: $9 + 6 = 15$

Final check: $15 = 15$ (This is true!)

So, $x = 13$ is indeed a solution.

Now, let's check $x = -5$:

Original equation: $|x - 4| + 6 = 15$

Substitute $x = -5$: $|-5 - 4| + 6 = 15$

Simplify: $|-9| + 6 = 15$

Further simplification: $9 + 6 = 15$

Final check: $15 = 15$ (This is also true!)

So, $x = -5$ is also a solution. Since both of our solutions check out, we can confidently say that they are the correct answers. Checking solutions is a valuable habit to cultivate in mathematics. It's a way to ensure that our calculations are accurate and that we haven't made any mistakes along the way. In the case of absolute value equations, checking is particularly important because the process of splitting the equation into two cases can sometimes introduce extraneous solutions. By plugging our solutions back into the original equation, we can verify that they are valid and that we haven't inadvertently included any values that don't actually satisfy the equation. So, always remember to check your solutions, it's the final piece of the puzzle!

The Answer

Okay, everyone, we've gone through the entire process step-by-step, and we've arrived at our answer! We found that the solutions to the equation $|x-4|+6=15$ are $x = 13$ and $x = -5$. So, the correct answer is B. $x=13$ and $x=-5$. Woohoo! We did it! This entire journey, from understanding the basics of absolute value to isolating the absolute value expression, setting up the two equations, solving them, and finally checking our solutions, has hopefully given you a solid grasp of how to tackle absolute value equations. Remember, the key is to break down the problem into smaller, manageable steps. Don't be intimidated by the absolute value bars; they're just a symbol that tells us to consider two possibilities. By following these steps, you'll be able to confidently solve a wide range of absolute value equations. And remember, practice makes perfect! The more you work through these types of problems, the more comfortable and confident you'll become. So, keep practicing, keep exploring, and keep challenging yourself. Math can be fun, and it's definitely a skill that you can master with dedication and the right approach. So, congratulations on making it to the end, and keep up the great work! You've got this! Now go out there and conquer some more math challenges!