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

Unveiling the Mystery: Understanding the Equation

Let's embark on a journey to solve the absolute value equation $-|-x| = -12$. This might look a bit intimidating at first glance, but fear not! We'll break it down into manageable steps, demystifying the process and revealing the solution set. Our primary goal is to determine the values of x that satisfy this equation. The presence of the absolute value bars introduces a unique twist, as we'll soon discover. This equation belongs to the realm of algebra, specifically dealing with absolute values. It's a fundamental concept in mathematics with applications in various fields. The absolute value of a number represents its distance from zero on the number line. Consequently, absolute values are always non-negative. Understanding this core principle is essential for solving absolute value equations accurately. Before we dive into the equation, let's refresh our understanding of the absolute value function. The absolute value of a number, denoted by |x|, is defined as: |x| = x if x ≥ 0 and |x| = -x if x < 0. This means that the absolute value of a number is always positive, regardless of the sign of the original number. For example, |-5| = 5 and |5| = 5. Now, back to our equation: $-|-x| = -12$. The negative sign outside the absolute value bars plays a crucial role. It means we are dealing with the negative of the absolute value of -x. We have to carefully consider how this negative sign impacts our calculations. The initial step in solving this equation involves isolating the absolute value term. This means getting the absolute value expression, which is $|-x|$, by itself on one side of the equation. This will allow us to simplify the equation and identify the possible values of x. Let's get started and unravel the mystery of this mathematical problem. The ultimate aim is to find the values of x that make the equation true. Let's delve in and uncover those x values that bring about the equation's truth. Our approach involves isolating the absolute value, then considering the two possibilities that the definition of absolute values opens up. This detailed approach provides us with the tools to solve this specific problem and other similar ones in the future. Now, let’s delve deeper into the core of the problem and unveil its elegant solution.

Step-by-Step Solution: Unpacking the Equation

Alright, let's roll up our sleeves and systematically solve the equation. The equation is $-|-x| = -12$. The first step is to isolate the absolute value term, which is $|-x|$. To do this, we need to get rid of that negative sign in front of the absolute value. We can achieve this by multiplying both sides of the equation by -1. Doing so gives us: $-1 * (-|-x|) = -1 * (-12)$. Which simplifies to $|-x| = 12$. Now we have successfully isolated the absolute value term. This is a crucial step towards finding the solution. The equation now reads $|-x| = 12$. This means the absolute value of -x is equal to 12. Remember, the absolute value of a number represents its distance from zero. Therefore, we have two possibilities to consider. The expression inside the absolute value bars, -x, can either be equal to 12 or -12. So, we set up two separate equations:

1. -x = 12
2. -x = -12.

Let's solve the first equation, -x = 12. To find x, we divide both sides by -1: x = -12. Now, let's solve the second equation, -x = -12. Again, we divide both sides by -1: x = 12. So we've found two possible solutions: x = -12 and x = 12. But are these our final answers? We must always verify our solutions. Verification is essential in solving absolute value equations to ensure we haven't made any errors. Let's substitute each value back into the original equation, $-|-x| = -12$, to verify their validity. First, let's try x = -12. Substituting this into the original equation, we get $-|-(-12)| = -12$, which simplifies to $-|12| = -12$, and finally to -12 = -12. This is true, meaning x = -12 is a valid solution. Next, let's try x = 12. Substituting this into the original equation, we get $-|-12| = -12$, which simplifies to $-12 = -12$. This is also true, meaning x = 12 is also a valid solution. We successfully went through the solution process. Now we have confirmed that both solutions satisfy the original equation. We've tackled the problem in an organized way. This methodical process allows us to unravel the equation step by step, which highlights the logic behind it.

Solution Set and Conclusion

After working through each step of the equation, we've successfully found the solution set. The solutions that satisfy the equation $-|-x| = -12$ are x = -12 and x = 12. Therefore, the solution set is -12, 12}. The key to solving this absolute value equation lies in understanding the definition of absolute value and how it impacts the equation. We isolated the absolute value term, considered both positive and negative possibilities, and verified our solutions. This approach can be applied to other absolute value equations, demonstrating the versatility of the method. The methodical approach we have taken here serves as a powerful tool in solving absolute value equations. Each step builds on the previous one, leading us to a clear and concise solution. In conclusion, the equation $-|-x| = -12$ has two solutions x = -12 and x = 12. These solutions belong to the set of real numbers and represent the values of x that make the original equation true. The process of solving an absolute value equation not only requires a strong grasp of mathematical principles, but also logical thinking. Practicing problems such as this enhances critical-thinking skills. This step-by-step approach serves as a guide for approaching similar problems. The solution set {-12, 12 completely describes the values of x for which the given absolute value equation holds true. Remember, consistency and careful attention to details are very important when solving mathematical problems. Congratulations on solving this equation! You've successfully navigated the world of absolute values. Keep practicing and keep exploring the endless world of math. You can now confidently tackle other absolute value equations, applying these concepts. We hope this comprehensive guide has been helpful and insightful. Now, you’re well-equipped to face similar problems with confidence and precision.

For further exploration and practice, you can explore resources like Khan Academy and other educational platforms, or consult a textbook on algebra. These resources can give you more problems and exercises.