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

Hey guys! Today, we're diving into a common math problem: solving equations that involve absolute values. Specifically, we're gonna tackle the equation $-6 + 3|4x - 3| = 0$. Don't worry, it might look a little intimidating at first, but I promise, we'll break it down into manageable steps. Understanding absolute values is super important, not just for your math class but also for building a strong foundation in algebra. So, let's get started and make sure you grasp every step. This guide will walk you through each stage of the process, from isolating the absolute value expression to finding the final solutions. We'll also look at what these solutions mean and how to check your answers. By the end of this, you'll be confident in solving similar problems. So, grab your pencils and let's get to work!

Understanding Absolute Value

Before we jump into the equation, let's quickly refresh what absolute value actually is. The absolute value of a number is its distance from zero on the number line. It's always a non-negative value. Think of it like this: no matter if you're walking 5 steps forward or 5 steps backward, the distance you've traveled is still 5 steps. We denote the absolute value of a number 'x' as |x|. For example, |-5| = 5 and |5| = 5. This concept is fundamental to understanding and solving the kind of equation we're dealing with.

This is because when we have an absolute value equation, we're essentially saying that the expression inside the absolute value bars can be either a positive or a negative value, since both would result in the same absolute value. For instance, if |x| = 3, then x could be 3 or -3. Understanding this dual possibility is key to correctly solving absolute value equations. Because of this, we'll often end up with two possible solutions. This is because the stuff inside the absolute value bars can equal the positive or negative version of the result after the absolute value part is isolated. That's why these problems are a bit more complex than simple linear equations. The good news is, once you get the hang of the process, they become pretty straightforward. Keep in mind the key principle: Absolute value measures distance from zero, and therefore, both positive and negative values within the absolute value bars can result in the same answer. Let's make sure to keep that in mind as we solve.

Step-by-Step Solution of $-6 + 3|4x - 3| = 0$

Alright, let's start solving the equation step-by-step. We are going to be super methodical so that you all can follow along. Here's the equation again: $-6 + 3|4x - 3| = 0$. We'll follow a structured approach to make sure we get the correct answers. It is important to know that our primary goal is to isolate the absolute value expression. The steps we will take will help us achieve this. Remember, isolating the absolute value means getting the |4x - 3| by itself on one side of the equation. The following steps will show you exactly how to do it.

Step 1: Isolate the Absolute Value Term

First, let's move all the terms that are not within the absolute value to the other side of the equation. In our case, we have the -6 and the 3 multiplying the absolute value. We need to get rid of these to isolate the absolute value term. To begin, add 6 to both sides of the equation. This cancels out the -6 on the left side, leaving us with:

$3|4x - 3| = 6$

Now, to completely isolate the absolute value, we need to get rid of the 3 that's multiplying the absolute value. We can do this by dividing both sides of the equation by 3. This gives us:

$|4x - 3| = 2$

Awesome! We've isolated the absolute value expression. This is a really important step, because now we can see the core of the problem: the absolute value of (4x - 3) equals 2. From this point, we can figure out what (4x - 3) is equal to.

Step 2: Set Up Two Separate Equations

Since the absolute value of an expression can be either positive or negative, we need to consider two possible cases:

• Case 1: The expression inside the absolute value is positive:

  $4x - 3 = 2$

• Case 2: The expression inside the absolute value is negative:

  $4x - 3 = -2$

This is a crucial step. It's where we account for both possible scenarios within the absolute value. We set up these two equations because, remember, the absolute value of a number is its distance from zero, so the expression inside could be either 2 or -2.

Step 3: Solve Each Equation

Now we have two simple linear equations to solve. Let's work through them one by one:

• Solving Case 1:

  $4x - 3 = 2$ Add 3 to both sides:

  $4x = 5$ Divide both sides by 4:

  x = rac{5}{4}

• Solving Case 2:

  $4x - 3 = -2$ Add 3 to both sides:

  $4x = 1$ Divide both sides by 4:

  x = rac{1}{4}

Boom! We've got two potential solutions: x = 5/4 and x = 1/4. But we are not done yet.

Step 4: Check Your Solutions

It is always a good idea to check our solutions by plugging them back into the original equation. This helps us to make sure we didn't make any mistakes along the way, and that our solutions are correct. Here's how we do it:

• Checking x = 5/4:

  -6 + 3|4(rac{5}{4}) - 3| = -6 + 3|5 - 3| = -6 + 3|2| = -6 + 3(2) = -6 + 6 = 0

  Yep, it checks out!

• Checking x = 1/4:

  -6 + 3|4(rac{1}{4}) - 3| = -6 + 3|1 - 3| = -6 + 3|-2| = -6 + 3(2) = -6 + 6 = 0

  Also checks out! We can have confidence that both of these answers are correct, as they make the original equation true. Doing these checks is a really important step in solving any math problem, because it helps you catch any errors and builds your confidence in your solution.

Conclusion

So, there you have it, guys! We have successfully solved the absolute value equation $-6 + 3|4x - 3| = 0$. By following our easy steps, we found that the solutions are x = 5/4 and x = 1/4. We made sure to check our work by plugging those values back into the original equation to verify our answer. Remember, solving absolute value equations involves isolating the absolute value, setting up two separate equations, solving them independently, and finally, checking your solutions. Keep practicing, and these problems will become second nature. Great job! You’ve added another tool to your mathematical toolkit. Until next time, keep those math muscles flexing!