Solve Absolute Value Equations: Find The Missing Solution

Aug 6, 2025 by Kenji Nakamura 58 views

Manuela's Equation Solution: Finding the Missing Piece

Hey guys! Let's dive into this math problem where Manuela's trying to solve an equation with absolute values. It looks like she's found one solution, but we need to figure out what the other one is. Absolute value equations can be a bit tricky, so we'll break it down step by step to make sure we get it right. We'll go through Manuela's work, spot where the absolute value plays its part, and then uncover that missing solution. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we fully grasp what we're dealing with. The equation Manuela is tackling is $3 - 2|0.5x + 1.5| = 2$ . The key here is the absolute value part, $|0.5x + 1.5|$ . Remember, absolute value means the distance from zero, so it's always non-negative. This is super important because it means that whatever's inside those absolute value bars, whether it's positive or negative, will come out as a positive value. This is the core concept we need to keep in mind as we work through this problem.

Now, let's think about what this means for finding solutions. Because of the absolute value, there are generally two possibilities we need to consider. For example, if $|a| = 5$ , then $a$ could be either 5 or -5, right? This is because both 5 and -5 are a distance of 5 away from zero. Similarly, in our equation, the expression inside the absolute value, $0.5x + 1.5$ , could be equal to 0.5 or -0.5. This is why we often end up with two solutions when dealing with absolute value equations. Manuela has found one solution, which means we need to hunt down the other one by considering the negative case.

So, to recap, the absolute value introduces a fork in the road. We need to consider both the positive and negative possibilities of the expression inside the absolute value bars to find all the solutions. This understanding is crucial as we move forward and analyze Manuela's steps and find the missing solution. Keep this in mind, and the rest will be a breeze!

Analyzing Manuela's Work

Alright, let's take a closer look at what Manuela did to solve the equation. Here are the steps she took:

$3 - 2|0.5x + 1.5| = 2$
$-2|0.5x + 1.5| = -1$
$|0.5x + 1.5| = 0.5$
$0.5x + 1.5 = 0.5$
$0.5x = -1$
$x = -2$

So far, so good! Manuela started by isolating the absolute value term, which is the correct approach. She subtracted 3 from both sides, divided by -2, and got to $|0.5x + 1.5| = 0.5$ . This is where the magic of absolute values really comes into play, and it's also the crucial step where we need to consider both possibilities. Up to this point, everything looks perfect, but it's important to recognize where the two paths diverge.

Manuela then took the positive case, which is $0.5x + 1.5 = 0.5$ . She solved this linearly, subtracting 1.5 from both sides and then dividing by 0.5, correctly finding the solution $x = -2$ . But remember, absolute value equations usually have two solutions because the expression inside the absolute value bars can be either positive or negative. Manuela only considered the positive case, and that's where we need to step in and explore the other possibility.

The key takeaway here is that while Manuela's algebraic manipulations were spot-on, she only explored one branch of the solution. To find the other solution, we need to consider what happens when the expression inside the absolute value bars is equal to the negative of the value on the right side of the equation. So, we're not faulting her work; we're just extending it to capture the complete picture. Next up, we'll tackle that missing negative case and find the other solution!

Finding the Other Solution

Okay, guys, let's get to the exciting part – finding that missing solution! We know from our analysis of Manuela's work that she correctly isolated the absolute value and considered the positive case. Now, we need to consider the negative case. Remember, when we have $|0.5x + 1.5| = 0.5$ , this means that the expression inside the absolute value, $0.5x + 1.5$ , can be either 0.5 or -0.5. Manuela already solved for the 0.5 case; let's tackle the -0.5 case.

So, we set up the equation:

$0.5x + 1.5 = -0.5$

Now, it's just a matter of solving for $x$ . First, we subtract 1.5 from both sides:

$0.5x = -0.5 - 1.5$

$0.5x = -2$

Then, we divide both sides by 0.5:

$x = -2 / 0.5$

$x = -4$

And there we have it! The other solution to the equation is $x = -4$ . Isn't it satisfying when you find that missing piece? By considering the negative case, we completed the solution set and found the second value of $x$ that makes the original equation true. This is a classic example of why absolute value equations are interesting – they often have two solutions, and it's our job to find them both. Next, we'll confirm our answer by plugging it back into the original equation to ensure everything checks out.

Verifying the Solution

Alright, let's put our detective hats back on and verify that $x = -4$ is indeed a solution to the original equation. This is a crucial step, guys, because it ensures that we haven't made any sneaky mistakes along the way. We'll plug $x = -4$ back into the original equation, $3 - 2|0.5x + 1.5| = 2$ , and see if it holds true.

So, here we go:

$3 - 2|0.5(-4) + 1.5| = 2$

First, let's simplify inside the absolute value:

$3 - 2|-2 + 1.5| = 2$

$3 - 2|-0.5| = 2$

Now, remember that the absolute value of -0.5 is 0.5:

$3 - 2(0.5) = 2$

Next, we perform the multiplication:

$3 - 1 = 2$

And finally, the subtraction:

$2 = 2$

Woohoo! It checks out! When we substitute $x = -4$ into the original equation, we get a true statement. This confirms that $x = -4$ is indeed a valid solution. Verification is always a good practice in mathematics, especially when dealing with absolute values, to make sure we haven't missed any subtle nuances or made any calculation errors. It gives us that extra confidence that we've solved the problem correctly. So, we've found the other solution, we've verified it, and we're feeling pretty good about our mathematical sleuthing skills!

Final Answer and Key Takeaways

Okay, guys, let's wrap things up! We started with Manuela's work on the equation $3 - 2|0.5x + 1.5| = 2$ , where she found one solution, $x = -2$ . Our mission was to find the other solution. By carefully analyzing Manuela's steps and understanding the nature of absolute value equations, we realized that we needed to consider both the positive and negative cases of the expression inside the absolute value bars.

We set up the equation for the negative case, $0.5x + 1.5 = -0.5$ , and solved for $x$ , which led us to the other solution, $x = -4$ . To be absolutely sure, we then verified our solution by plugging $x = -4$ back into the original equation, and it checked out perfectly.

Therefore, the other solution to the equation is $x = -4$ . This corresponds to option B.

So, what are the key takeaways from this problem? First, remember that absolute value equations often have two solutions because the expression inside the absolute value can be either positive or negative. Second, always isolate the absolute value term before splitting the problem into two cases. Third, and perhaps most importantly, always verify your solutions by plugging them back into the original equation. This helps prevent errors and ensures that you have the correct answers. And finally, don't be afraid to break down the problem step by step, just like we did, to make sure you understand each step and don't miss any details. Great job, everyone! Keep practicing, and you'll become absolute value equation masters in no time!