Solving The Absolute Value Equation: $-2|2.2x-3.3|=-6.6$

Hey Plastik Magazine readers! Let's dive into solving an absolute value equation today. We've got a fun one: $-2|2.2x-3.3|=-6.6$. If you're scratching your head, don't worry, we'll break it down step by step. Whether you're a math whiz or just trying to brush up on your algebra skills, this guide will help you tackle this problem with confidence. So, grab your pencils and let's get started!

Understanding Absolute Value

Before we jump into the equation itself, let’s quickly recap what absolute value means. Absolute value is the distance of a number from zero on the number line. It’s always non-negative. Think of it like this: $|5|$ is 5, and $|-5|$ is also 5. The absolute value strips away the sign, leaving only the magnitude. This concept is crucial for solving equations like ours, where we have an absolute value expression.

When we're dealing with equations containing absolute values, we need to consider two possibilities: the expression inside the absolute value can be either positive or negative. This is because both the positive and negative versions of a number have the same absolute value. For instance, if $|x| = 3$, then x could be either 3 or -3. This principle will guide us as we solve our equation.

Remember, the key takeaway here is that absolute value represents distance from zero. This means we have to account for both positive and negative scenarios when solving equations. With that in mind, let’s move on to the next step and isolate the absolute value term in our equation.

Isolating the Absolute Value

Alright, first things first, we need to isolate the absolute value part of the equation. In our case, that's the $|2.2x-3.3|$ term. The equation we're working with is $-2|2.2x-3.3|=-6.6$. To get the absolute value by itself, we need to get rid of that -2 that’s hanging out in front. How do we do that? Simple! We divide both sides of the equation by -2.

So, we have:

$$\frac{-2|2.2x-3.3|}{-2} = \frac{-6.6}{-2}$$

This simplifies to:

$$|2.2x-3.3| = 3.3$$

Now we're talking! We've successfully isolated the absolute value. This is a crucial step because it sets us up to handle the two possibilities we discussed earlier. By getting the absolute value term alone, we can now consider both the positive and negative scenarios for the expression inside the absolute value bars. This is where the fun really begins!

Think of it like clearing the stage before a big performance. We've removed the distractions and now we can focus on the main act: solving for x. With the absolute value isolated, we’re ready to split the equation into two separate cases and find the solutions that satisfy the original equation.

Splitting into Two Cases

Okay, now that we've isolated the absolute value, it's time to tackle the heart of the problem. Remember how we said absolute value means we need to consider both positive and negative possibilities? This is where that comes into play. Since $|2.2x-3.3| = 3.3$, the expression inside the absolute value, 2.2x - 3.3, can be either 3.3 or -3.3. This gives us two separate equations to solve.

Case 1: The positive case

In the first case, we assume that the expression inside the absolute value is equal to the positive value:

$$2.2x - 3.3 = 3.3$$

This is a straightforward linear equation. We'll solve it in the next step. But first, let's set up the second case.

Case 2: The negative case

In the second case, we assume that the expression inside the absolute value is equal to the negative value:

$$2.2x - 3.3 = -3.3$$

Again, this is a linear equation that we can solve. By splitting the absolute value equation into these two cases, we’ve transformed a single, slightly tricky equation into two simpler ones. This is a common strategy when dealing with absolute values, and it makes the problem much more manageable.

Think of it like having a fork in the road. We've identified the two possible paths and now we need to explore each one separately to find our destinations (the solutions for x). Let’s move on and solve these two equations.

Solving Case 1: 2.2x - 3.3 = 3.3

Let’s dive into solving the first case: $2.2x - 3.3 = 3.3$. This is a classic linear equation, and we'll solve it using standard algebraic techniques. Our goal is to isolate x on one side of the equation. The first step is to get rid of that -3.3. We do this by adding 3.3 to both sides of the equation.

So, we have:

$$2.2x - 3.3 + 3.3 = 3.3 + 3.3$$

This simplifies to:

$$2.2x = 6.6$$

Great! Now, x is almost by itself. We just need to get rid of the 2.2 that's multiplying it. We do this by dividing both sides of the equation by 2.2:

$$\frac{2.2x}{2.2} = \frac{6.6}{2.2}$$

This simplifies to:

$$x = 3$$

Boom! We've found our first solution. x = 3 is one value that satisfies the original absolute value equation. But remember, we have two cases to consider. So, we’re not done yet. We still need to solve the second case to see if there’s another solution lurking out there.

Solving linear equations like this is a fundamental skill in algebra. By carefully applying the same operation to both sides of the equation, we maintain the balance and move closer to isolating the variable. Now, let's tackle the second case and see what it holds.

Solving Case 2: 2.2x - 3.3 = -3.3

Now, let's tackle the second case: $2.2x - 3.3 = -3.3$. Just like before, we're dealing with a linear equation, and our mission is to isolate x. We'll follow the same steps as in Case 1, starting by adding 3.3 to both sides of the equation. This will cancel out the -3.3 on the left side.

So, we have:

$$2.2x - 3.3 + 3.3 = -3.3 + 3.3$$

This simplifies to:

$$2.2x = 0$$

Alright, things are looking interesting! Now, to get x completely alone, we need to divide both sides of the equation by 2.2:

$$\frac{2.2x}{2.2} = \frac{0}{2.2}$$

This simplifies to:

$$x = 0$$

Awesome! We've found our second solution. x = 0 also satisfies the original absolute value equation. So, it looks like we have two solutions in total.

This case demonstrates a slightly different scenario than the first. Here, the constant terms canceled each other out, leading to a solution of x = 0. This is a perfectly valid outcome and highlights the importance of carefully working through each step of the equation.

Checking the Solutions

Before we declare victory, it's always a good idea to check our solutions. This is a crucial step to make sure we haven't made any mistakes along the way. We have two solutions: x = 3 and x = 0. Let’s plug each one back into the original equation, $-2|2.2x-3.3|=-6.6$, and see if they hold true.

Checking x = 3

Substitute x = 3 into the equation:

$$-2|2.2(3)-3.3| = -6.6$$

Simplify the expression inside the absolute value:

$$-2|6.6-3.3| = -6.6$$

$$-2|3.3| = -6.6$$

$$-2(3.3) = -6.6$$

$$-6.6 = -6.6$$

It checks out! x = 3 is indeed a valid solution.

Checking x = 0

Now, let's substitute x = 0 into the equation:

$$-2|2.2(0)-3.3| = -6.6$$

Simplify the expression inside the absolute value:

$$-2|0-3.3| = -6.6$$

$$-2|-3.3| = -6.6$$

$$-2(3.3) = -6.6$$

$$-6.6 = -6.6$$

It checks out too! x = 0 is also a valid solution.

By plugging our solutions back into the original equation, we've confirmed that both x = 3 and x = 0 are correct. This process of verification is an essential part of problem-solving in mathematics. It gives us confidence in our answers and helps us catch any potential errors.

Final Answer

Alright guys, we've reached the finish line! We've successfully solved the absolute value equation $-2|2.2x-3.3|=-6.6$. We isolated the absolute value, split the equation into two cases, solved each case, and even checked our solutions. Phew! That's a lot of mathing!

So, what's the final answer? We found two solutions:

• x = 3
• x = 0

Therefore, the correct answer is D. x = 0 or x = 3.

Absolute value equations might seem intimidating at first, but by breaking them down into manageable steps, they become much less scary. Remember the key concepts: absolute value represents distance from zero, and we need to consider both positive and negative possibilities. Keep practicing, and you'll become a pro at solving these types of equations in no time!