Solving $-4|2x-3|+18=6$: A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of absolute value equations. Specifically, we're going to tackle the equation $-4|2x-3|+18=6$. Don't worry, it might look a bit intimidating at first, but we'll break it down step-by-step so you can conquer these types of problems with confidence. So, grab your pencils, notebooks, and let's get started!

Understanding Absolute Value

Before we jump into the nitty-gritty of solving the equation, let's quickly refresh our understanding of absolute value. Absolute value, at its core, represents the distance of a number from zero on the number line. This distance is always non-negative. Think of it like this: whether you walk 5 steps to the right (positive 5) or 5 steps to the left (negative 5) from zero, you've still traveled a distance of 5 steps. Mathematically, we denote the absolute value of a number 'x' as |x|. So, |5| = 5 and |-5| = 5. This concept is crucial because it means that inside an absolute value, we have to consider two possibilities: the expression inside could be positive or negative, and both scenarios lead to a valid solution.

When dealing with absolute value equations, like the one we are about to solve, understanding this dual nature is paramount. The equation $-4|2x-3|+18=6$ contains the absolute value expression $|2x-3|$. This means that the expression '2x-3' could be either positive or negative, and we need to consider both cases to find all possible solutions for 'x'. Ignoring this dual possibility is a common mistake, so always remember to split the problem into two separate equations based on the two potential scenarios within the absolute value bars. By doing so, you ensure that you capture all possible values of 'x' that satisfy the original equation. This methodical approach is the key to accurately solving absolute value equations.

The absolute value function essentially strips away the sign of a number, leaving us with its magnitude. This seemingly simple operation has profound implications when solving equations. It introduces a branching path in our solution process, compelling us to address two distinct possibilities arising from the absolute value expression. Let's consider a simple example: |x| = 3. This equation implies that x could be either 3 or -3, because both these numbers are 3 units away from zero. Similarly, in our given equation, $-4|2x-3|+18=6$, the expression inside the absolute value, '2x-3', could result in a positive value or a negative value that, after taking the absolute value and performing the arithmetic operations, yields the final result. Therefore, we must meticulously explore both scenarios to uncover the complete solution set.

Isolating the Absolute Value

Our first major step in solving $-4|2x-3|+18=6$ is to isolate the absolute value term. Think of it like peeling an onion – we need to get to the core, which in this case is the absolute value expression itself. To do this, we'll use the basic principles of algebra to get the |2x-3| by itself on one side of the equation. Remember, our goal is to perform operations on both sides of the equation to maintain balance while simplifying it. This isolation step is absolutely critical because it sets the stage for the next crucial step: splitting the equation into two separate cases.

The initial equation is $-4|2x-3|+18=6$. To isolate the absolute value, we first need to get rid of the '+18' term. We can do this by subtracting 18 from both sides of the equation. This gives us: $-4|2x-3| = 6 - 18$, which simplifies to $-4|2x-3| = -12$. Great! We've taken the first step towards isolating our absolute value term. Now, we need to deal with the '-4' that's multiplying the absolute value. To undo this multiplication, we divide both sides of the equation by -4. This results in: $|2x-3| = -12 / -4$, which simplifies to $|2x-3| = 3$. And there you have it! We've successfully isolated the absolute value expression. This is a significant milestone in solving the equation.

By meticulously isolating the absolute value term, we've essentially prepared the equation for the next crucial phase: considering the two possible scenarios arising from the absolute value. Isolating the absolute value term is not just a procedural step; it's a strategic move that allows us to clearly see the core of the problem. Once the absolute value is isolated, we can confidently apply the definition of absolute value and split the equation into two separate cases, each representing a different possibility for the expression inside the absolute value. This methodical approach ensures that we don't miss any potential solutions and that we tackle the problem in a systematic and organized manner. Remember, a well-isolated absolute value expression is the gateway to solving the equation effectively and accurately.

Splitting into Two Cases

Now comes the crucial step where we leverage our understanding of absolute value: splitting the equation into two separate cases. Remember, the absolute value of an expression is its distance from zero, which means the expression inside the absolute value bars could be either positive or negative. In our equation, $|2x-3| = 3$, this means that either 2x-3 = 3 or 2x-3 = -3. We need to consider both of these possibilities to find all possible solutions for 'x'. This is where many people make mistakes, so pay close attention!

Case 1: The expression inside the absolute value is positive or zero. This means 2x - 3 = 3. We'll solve this equation for 'x' using basic algebraic techniques. First, we add 3 to both sides of the equation: 2x = 3 + 3, which simplifies to 2x = 6. Then, we divide both sides by 2: x = 6 / 2, which gives us x = 3. So, one potential solution is x = 3. But don't stop there! We still have another case to consider.

Case 2: The expression inside the absolute value is negative. This means 2x - 3 = -3. Again, we'll solve this equation for 'x'. First, we add 3 to both sides: 2x = -3 + 3, which simplifies to 2x = 0. Then, we divide both sides by 2: x = 0 / 2, which gives us x = 0. So, our second potential solution is x = 0. By splitting the equation into these two distinct cases, we've ensured that we've captured all possible scenarios and haven't missed any solutions. This methodical approach is the key to solving absolute value equations accurately.

Splitting the absolute value equation into two separate cases is not just a mathematical technique; it's a reflection of the fundamental nature of the absolute value function. By acknowledging that the expression inside the absolute value bars can be either positive or negative, we embark on a comprehensive exploration of all possible scenarios. This meticulous approach is essential for solving these types of equations correctly. Failing to consider both cases can lead to incomplete solutions, which means missing valid answers. Therefore, always remember to split the equation and solve each case independently. This strategic step is the cornerstone of solving absolute value equations accurately and effectively.

Solving Each Case

Now that we've split our equation into two cases, let's solve each one individually. This is where our basic algebra skills come into play. We'll use the familiar steps of adding, subtracting, multiplying, and dividing to isolate 'x' in each equation. Remember, the goal is to get 'x' by itself on one side of the equation so we can determine its value. Let's tackle each case one at a time.

In Case 1, we have the equation 2x - 3 = 3. To solve for 'x', we first add 3 to both sides: 2x = 3 + 3, which simplifies to 2x = 6. Next, we divide both sides by 2: x = 6 / 2, which gives us our first solution, x = 3. So, in the first case, we found that 'x' equals 3.

In Case 2, we have the equation 2x - 3 = -3. Again, we start by adding 3 to both sides: 2x = -3 + 3, which simplifies to 2x = 0. Then, we divide both sides by 2: x = 0 / 2, which gives us our second solution, x = 0. So, in the second case, we found that 'x' equals 0. By systematically solving each case, we've identified two potential solutions for our original equation. The methodical approach ensures that we've considered all possibilities and haven't overlooked any valid solutions.

Solving each case individually is a critical step in the process of solving absolute value equations. This approach ensures that we address all possibilities arising from the absolute value expression. Each case represents a different scenario for the expression inside the absolute value bars, and solving each case independently allows us to determine the potential values of the variable that satisfy the original equation. By methodically applying algebraic techniques to each case, we can isolate the variable and find its value, leading us to the complete solution set. Remember, the key is to treat each case as a separate equation and solve it using the standard rules of algebra. This diligent approach is essential for accurately solving absolute value equations.

Checking for Extraneous Solutions

This is a super important step that many people skip, but it can save you from getting the wrong answer! When dealing with absolute value equations, it's crucial to check your solutions for extraneous solutions. Extraneous solutions are values that you obtain during the solving process that don't actually satisfy the original equation. They can arise due to the nature of absolute value and the way we split the equation into cases. So, to be absolutely sure our solutions are correct, we need to plug them back into the original equation and see if they hold true.

Let's start by checking our first potential solution, x = 3. We plug this value into the original equation, $-4|2x-3|+18=6$: $-4|2(3)-3|+18=6$. Simplifying the expression inside the absolute value: $-4|6-3|+18=6$, which becomes $-4|3|+18=6$. Now, we take the absolute value: $-4(3)+18=6$, which simplifies to $-12+18=6$. Finally, we have $6=6$. This is a true statement, so x = 3 is indeed a valid solution. Great! Now, let's check our second potential solution, x = 0.

We plug x = 0 into the original equation: $-4|2(0)-3|+18=6$. Simplifying the expression inside the absolute value: $-4|0-3|+18=6$, which becomes $-4|-3|+18=6$. Now, we take the absolute value: $-4(3)+18=6$, which simplifies to $-12+18=6$. Again, we have $6=6$. This is also a true statement, so x = 0 is a valid solution as well. By checking both solutions, we've confirmed that they both satisfy the original equation and are not extraneous. This thoroughness is key to ensuring the accuracy of our solutions.

Checking for extraneous solutions is not just a procedural step; it's a safeguard against potential errors. This practice allows us to verify whether the solutions obtained during the solving process are genuine solutions of the original equation. Extraneous solutions often arise due to the manipulation of the equation during the solving process, particularly when dealing with absolute values or radicals. By substituting the potential solutions back into the original equation, we can determine whether they make the equation true or lead to a contradiction. This step provides a crucial validation of our work and ensures that we present only the correct solutions. So, always remember to check for extraneous solutions to maintain the integrity of your solution process.

The Solution Set

After all that hard work, we've arrived at the final step: stating the solution set. We've solved the equation, considered both cases, and checked for extraneous solutions. Now, we simply need to express our solutions in a clear and concise way. The solution set is the collection of all values that satisfy the original equation. In our case, we found two solutions: x = 3 and x = 0. Therefore, our solution set consists of these two values.

We can express the solution set in a couple of ways. One common way is to use set notation, which involves listing the solutions within curly braces: {0, 3}. This notation clearly indicates that the solutions to the equation are 0 and 3. Another way to express the solution set is to simply state: