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

Hey guys! Let's dive into solving absolute value equations. We're going to break down the equation: $\left|\frac{x}{6}-\frac{7}{12}\right|=\frac{11}{6}$. Don't worry, it might look a little intimidating at first, but trust me, it's totally manageable once you get the hang of it. We'll go through it step-by-step, making sure you understand every part of the process. Absolute value equations are like puzzles, and we're here to solve them together! This guide will provide you with the necessary tools and knowledge to confidently tackle these kinds of problems, making you a pro at dealing with absolute values. We will start with a clear understanding of the absolute value concept, followed by a detailed walkthrough of the given equation. Throughout this journey, we'll keep the explanations simple and friendly, so you won't feel lost. By the end, you'll be able to solve similar equations with ease. Let's get started and make solving absolute value equations a piece of cake!

Understanding Absolute Value: The Foundation

Before we jump into the equation, let's make sure we're all on the same page about what absolute value actually is. Think of it like this: the absolute value of a number is its distance from zero on the number line. It doesn't matter if the number is positive or negative; distance is always positive. 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. So basically, the absolute value function makes everything positive. It's super important to remember that because it's the heart of solving these types of equations. Understanding this concept is the key to unlocking the puzzle. Now, why is this important? Because when you have an absolute value equation, you have to consider two possibilities: the expression inside the absolute value can be either positive or negative. Both scenarios can result in the same absolute value. This is the core principle that we will use to solve our equation. It is also important to remember that the absolute value of a number is always non-negative. It's always either zero or a positive number. In mathematical terms, the absolute value of a number x, denoted as |x|, is defined as: |x| = x, if x ≥ 0 and |x| = -x, if x < 0. So, to solve an absolute value equation, we consider both possibilities to find all possible solutions. Keep this in mind, and you will find these equations much easier to handle. Now that we understand the core concept, we're ready to move on.

Step-by-Step Solution: Cracking the Equation

Alright, let's get down to business and solve the equation $\left|\frac{x}{6}-\frac{7}{12}\right|=\frac{11}{6}$. As mentioned, the absolute value equation gives us two possibilities. First, the expression inside the absolute value signs is positive, and second, it's negative. Here's how we handle those possibilities:

Case 1: The Expression is Positive

If $\frac{x}{6}-\frac{7}{12}$ is positive, then we can just drop the absolute value signs and solve the equation as is. This gives us:

$\frac{x}{6}-\frac{7}{12}=\frac{11}{6}$

Now, let's solve for x. First, we add $\frac{7}{12}$ to both sides of the equation:

$\frac{x}{6}=\frac{11}{6}+\frac{7}{12}$

To add the fractions on the right side, we need a common denominator, which is 12. So, we rewrite $\frac{11}{6}$ as $\frac{22}{12}$: Then we get:

$\frac{x}{6}=\frac{22}{12}+\frac{7}{12}$

Now we can add the fractions:

$\frac{x}{6}=\frac{29}{12}$

Next, to isolate x, we multiply both sides of the equation by 6:

$x = \frac{29}{12} \times 6$

Simplify the right side: $x = \frac{29}{2}$ or x = 14.5. So, in this first case, we find that x = 14.5 is a possible solution.

Case 2: The Expression is Negative

If $\frac{x}{6}-\frac{7}{12}$ is negative, then the absolute value makes it positive. In this case, we have to flip the sign of the right side to get rid of the absolute value, that is, we take the negative of the expression inside the absolute value signs and solve for x. This gives us:

$\frac{x}{6}-\frac{7}{12}=-\frac{11}{6}$

Now, let's solve for x. Again, we add $\frac{7}{12}$ to both sides of the equation:

$\frac{x}{6}=-\frac{11}{6}+\frac{7}{12}$

Rewrite -$\frac{11}{6}$ as -$\frac{22}{12}$:

$\frac{x}{6}=-\frac{22}{12}+\frac{7}{12}$

Now we can add the fractions:

$\frac{x}{6}=-\frac{15}{12}$

To isolate x, we multiply both sides of the equation by 6:

$x = -\frac{15}{12} \times 6$

Simplify the right side: $x = -\frac{15}{2}$ or $x = -7.5$. Therefore, in the second case, we find that x = -7.5 is also a possible solution. Now we've got our two possible solutions!

Checking the Solutions: The Final Test

Before we say we're done, we need to make sure our solutions actually work. It's always a good idea to check your answers in the original equation to ensure there are no mistakes. We're going to plug each solution back into the original equation and verify that both sides are equal.

Checking x = 14.5

Let's substitute x = 14.5 or x = $\frac{29}{2}$ into the original equation $\left|\frac{x}{6}-\frac{7}{12}\right|=\frac{11}{6}$: Then we get:

$\left|\frac{14.5}{6}-\frac{7}{12}\right|=\frac{11}{6}$

Calculate the expression inside the absolute value: $\frac{14.5}{6} = \frac{29}{12}$. Then we have:

$\left|\frac{29}{12}-\frac{7}{12}\right|=\frac{11}{6}$

$\left|\frac{22}{12}\right|=\frac{11}{6}$

$\frac{22}{12}=\frac{11}{6}$ after simplification.

$\frac{11}{6}=\frac{11}{6}$, which is true. Therefore, x = 14.5 is a valid solution.

Checking x = -7.5

Now, let's substitute x = -7.5 or x = -$\frac{15}{2}$ into the original equation $\left|\frac{x}{6}-\frac{7}{12}\right|=\frac{11}{6}$: Then we get:

$\left|\frac{-7.5}{6}-\frac{7}{12}\right|=\frac{11}{6}$

Calculate the expression inside the absolute value: $\frac{-7.5}{6} = -\frac{15}{12}$. Then we have:

$\left|-\frac{15}{12}-\frac{7}{12}\right|=\frac{11}{6}$

$\left|-\frac{22}{12}\right|=\frac{11}{6}$

$\frac{22}{12}=\frac{11}{6}$ after simplification.

$\frac{11}{6}=\frac{11}{6}$, which is true. Therefore, x = -7.5 is a valid solution. We have verified both our solutions; we can be confident that these solutions are correct.

Conclusion: You've Got This!

Awesome work, guys! We've successfully solved the absolute value equation $\left|\frac{x}{6}-\frac{7}{12}\right|=\frac{11}{6}$! We found two solutions: x = 14.5 and x = -7.5. Remember that solving these types of equations is all about understanding the definition of absolute value and considering both positive and negative cases. Always double-check your answers to make sure they're correct. Keep practicing, and you'll become a pro in no time. If you have similar equations to solve, apply the same approach, and you'll be able to work it out. Good luck, and keep up the great work! You've taken a significant step toward mastering absolute value equations. Remember, the more you practice, the easier it becomes. So, keep at it, and soon you'll be solving these equations without any problems! Absolute value equations are a fundamental part of algebra, and by conquering this skill, you've opened the door to many more mathematical adventures. Keep practicing, and you'll do great!