Solving -abs(-x) = -12 Solution Set Explained

JU07/09/2025 08, 2025 by THE IDEN 46 views

Unraveling the Solution Set of -|-x| = -12: A Comprehensive Guide

The realm of mathematics often presents us with intriguing equations that require careful analysis and a systematic approach to decipher their solutions. One such equation is -|-x| = -12, which involves the absolute value function and its interaction with negative signs. In this comprehensive guide, we will embark on a journey to unravel the solution set of this equation, providing a step-by-step explanation that caters to both novice learners and seasoned math enthusiasts. We will delve into the fundamental concepts of absolute value, explore the implications of negative signs, and ultimately arrive at the solution set with clarity and precision.

Understanding Absolute Value: The Foundation of Our Exploration

At the heart of our equation lies the absolute value function, denoted by the vertical bars | |. The absolute value of a number represents its distance from zero on the number line, regardless of its sign. In simpler terms, it transforms any negative number into its positive counterpart while leaving positive numbers unchanged. For instance, |5| = 5 and |-5| = 5. This inherent property of the absolute value function plays a crucial role in shaping the solutions of equations that involve it.

To grasp the essence of absolute value, let's consider a few more examples. The absolute value of 0, denoted as |0|, is simply 0, as it lies at the origin of the number line. Similarly, the absolute value of -10, denoted as |-10|, is 10, representing its distance of 10 units from zero. Understanding this fundamental concept is paramount to tackling equations involving absolute values.

Deciphering the Equation -|-x| = -12: A Step-by-Step Approach

Now, let's turn our attention to the equation at hand: -|-x| = -12. This equation presents a unique challenge due to the presence of both the absolute value function and a negative sign. To solve it effectively, we will employ a systematic approach that involves isolating the absolute value term and then considering the implications of its definition.

Step 1: Isolating the Absolute Value Term

The first step in our journey is to isolate the absolute value term, |-x|, on one side of the equation. To achieve this, we can multiply both sides of the equation by -1. This operation effectively cancels out the negative sign in front of the absolute value term, giving us:

|-x| = 12

This seemingly simple step is crucial as it allows us to focus solely on the absolute value term and its relationship to the constant value on the right side of the equation.

Step 2: Applying the Definition of Absolute Value

Now that we have isolated the absolute value term, we can invoke the definition of absolute value to analyze the equation further. Recall that the absolute value of a number represents its distance from zero. Therefore, the equation |-x| = 12 implies that the distance between -x and zero is 12 units.

This leads us to consider two distinct possibilities: either -x is equal to 12 or -x is equal to -12. In the first case, if -x = 12, then multiplying both sides by -1 gives us x = -12. In the second case, if -x = -12, then multiplying both sides by -1 gives us x = 12.

Step 3: Identifying the Solution Set

From our analysis, we have arrived at two potential solutions: x = -12 and x = 12. To express the solution set concisely, we can use set notation, which represents the collection of all values that satisfy the equation. Therefore, the solution set of the equation -|-x| = -12 is {-12, 12}.

This solution set signifies that both -12 and 12 are valid values for x that make the equation true. We can verify this by substituting each value back into the original equation and confirming that the equality holds.

Delving Deeper: Visualizing the Solution on the Number Line

To further solidify our understanding of the solution set, let's visualize it on the number line. The number line provides a visual representation of numbers and their relationships, allowing us to grasp the concept of distance and absolute value more intuitively.

On the number line, we can mark the two solutions, -12 and 12, as distinct points. These points represent the values of x that satisfy the equation -|-x| = -12. The distance between each point and zero is precisely 12 units, which aligns with the definition of absolute value.

This visual representation reinforces the idea that the absolute value function captures the distance from zero, regardless of the direction. The solution set {-12, 12} encompasses all the points on the number line that are exactly 12 units away from zero.

Extending Our Knowledge: Exploring Variations of the Equation

Now that we have mastered the solution of -|-x| = -12, let's extend our knowledge by exploring variations of this equation. By modifying the coefficients and constants, we can gain a deeper appreciation for the interplay between absolute value and negative signs.

Consider the equation -2|-x| = -24. In this case, the absolute value term is multiplied by -2. To solve this equation, we can first divide both sides by -2, which gives us |-x| = 12. This equation is identical to the one we solved earlier, and its solution set remains {-12, 12}.

Another variation could be -|-x + 3| = -12. Here, the expression inside the absolute value is x + 3. To solve this equation, we can again multiply both sides by -1, resulting in |-x + 3| = 12. This equation implies that the distance between -x + 3 and zero is 12 units. We can then consider the two cases: -x + 3 = 12 and -x + 3 = -12, and solve for x accordingly.

By exploring these variations, we develop a more robust understanding of absolute value equations and their solutions.

Conclusion: Mastering the Art of Solving Absolute Value Equations

In this comprehensive guide, we have embarked on a journey to unravel the solution set of the equation -|-x| = -12. We began by understanding the fundamental concept of absolute value, its definition, and its representation on the number line. We then systematically solved the equation, isolating the absolute value term, applying its definition, and identifying the solution set {-12, 12}.

Furthermore, we delved deeper into the topic by visualizing the solution on the number line and exploring variations of the equation. Through this exploration, we have gained a profound understanding of absolute value equations and their solutions. Armed with this knowledge, you are now well-equipped to tackle a wide range of absolute value equations with confidence and precision.

The key to mastering absolute value equations lies in understanding the definition of absolute value, applying it systematically, and visualizing the solutions on the number line. By embracing these principles, you can unlock the power of absolute value and excel in your mathematical endeavors.

Solving mathematical equations can sometimes feel like deciphering a complex code. The equation -|-x|=-12 might appear daunting at first glance, but with a systematic approach and a firm understanding of the underlying principles, we can unlock its secrets and determine its solution set. This exploration will guide you through the process, breaking down the equation into manageable steps and illuminating the logic behind each manipulation. We will delve into the concept of absolute value, its properties, and how it interacts with negative signs. By the end of this journey, you'll not only understand the solution to this specific equation but also gain valuable insights into solving similar mathematical challenges.

Unveiling the Absolute Value: A Core Concept

The absolute value is a fundamental concept in mathematics that represents the distance of a number from zero on the number line. It essentially strips away the sign of a number, leaving only its magnitude. For instance, the absolute value of 5, denoted as |5|, is 5, and the absolute value of -5, denoted as |-5|, is also 5. This concept is crucial for understanding equations involving absolute values, as it introduces the possibility of multiple solutions.

Imagine a number line stretching infinitely in both positive and negative directions. The absolute value of a number is simply its distance from the central point, zero. Whether the number is to the left (negative) or to the right (positive) of zero, its absolute value is always a non-negative quantity. This non-negativity is a key characteristic of the absolute value function and plays a vital role in solving equations.

Navigating the Negatives: Understanding the Impact

The equation -|-x|=-12 features not one, but two negative signs. One is directly associated with the absolute value term, and the other is on the right side of the equation. Understanding how these negative signs interact is essential for solving the equation correctly. The negative sign outside the absolute value effectively inverts the result of the absolute value operation, while the negative sign on the right side sets the target value for the expression on the left.

Consider the expression -|-x|. The absolute value |-x| will always yield a non-negative value, as discussed earlier. However, the negative sign in front of the absolute value flips this non-negative value into its negative counterpart. This means that the entire expression -|-x| will always be non-positive (either negative or zero). This observation is crucial for understanding the constraints on the possible values of x.

Step-by-Step Solution: Deciphering the Equation -|-x|=-12

Now, let's embark on a step-by-step journey to solve the equation -|-x|=-12. We'll break down the process into clear, manageable steps, explaining the logic behind each manipulation.

Step 1: Isolate the Absolute Value

The first step is to isolate the absolute value term |-x| on one side of the equation. To do this, we can multiply both sides of the equation by -1. This operation cancels out the negative sign in front of the absolute value, giving us:

|-x| = 12

This step simplifies the equation, allowing us to focus solely on the absolute value term and its relationship to the constant value on the right side.

Step 2: Apply the Definition of Absolute Value

Now that we've isolated the absolute value, we can apply its definition to understand the implications of the equation. Recall that the absolute value of a number represents its distance from zero. The equation |-x| = 12 tells us that the distance between -x and zero is 12 units.

This leads to two possibilities: -x can be either 12 or -12. In other words, there are two numbers whose absolute value is 12: 12 itself and -12. This is a fundamental consequence of the absolute value function's nature.

Step 3: Solve for x

We now have two separate equations to solve:

-x = 12

and

-x = -12

To solve for x in each case, we can multiply both sides of each equation by -1. This gives us:

x = -12

and

x = 12

Step 4: Express the Solution Set

We've arrived at two potential solutions for x: -12 and 12. The solution set is the collection of all values that satisfy the equation. We can express this solution set using set notation:

{-12, 12}

This set contains two elements, -12 and 12, indicating that both of these values are valid solutions to the equation -|-x|=-12.

Visualizing the Solution: The Number Line Perspective

To gain a deeper understanding of the solution set, let's visualize it on the number line. Draw a number line extending from negative infinity to positive infinity, with zero at the center. Mark the points -12 and 12 on the number line.

You'll notice that both -12 and 12 are equidistant from zero, each being 12 units away. This visual representation reinforces the concept of absolute value as distance from zero. The solution set {-12, 12} represents all the points on the number line that are exactly 12 units away from the origin.

Expanding Our Horizons: Exploring Similar Equations

Now that we've mastered the solution to -|-x|=-12, let's broaden our knowledge by exploring similar equations. This will help you develop a more robust understanding of absolute value equations and their solutions.

Consider the equation -|-x+2|=-5. This equation introduces a slight variation by adding a constant term inside the absolute value. The solution process remains similar, but we need to account for the added constant.

First, multiply both sides by -1 to isolate the absolute value: |-x+2|=5. Then, apply the definition of absolute value to create two equations: -x+2=5 and -x+2=-5. Solve each equation for x to find the solution set.

By exploring variations like this, you'll develop a more flexible and adaptable approach to solving absolute value equations.

Conclusion: Mastering the Equation -|-x|=-12 and Beyond

In this comprehensive exploration, we've successfully decoded the equation -|-x|=-12 and determined its solution set {-12, 12}. We've delved into the fundamental concepts of absolute value, understood the impact of negative signs, and followed a step-by-step solution process. We've also visualized the solution on the number line and expanded our horizons by exploring similar equations.

By mastering this equation, you've gained valuable skills and insights that will empower you to tackle a wide range of mathematical challenges involving absolute values. Remember the key principles: understand the definition of absolute value, carefully handle negative signs, and systematically solve the resulting equations. With practice and perseverance, you'll become a confident and proficient problem solver.

Mathematical equations can sometimes seem like cryptic puzzles, but with the right tools and a methodical approach, we can decipher their hidden meanings and find their solutions. The equation -|-x| = -12 is a perfect example. It involves the absolute value function and negative signs, which can initially seem perplexing. However, by breaking it down step by step and understanding the underlying concepts, we can confidently determine its solution set. This guide will walk you through the process, explaining each step in detail and providing insights into the mathematical principles involved. Whether you're a student learning about absolute values or simply a curious mind seeking a deeper understanding, this exploration will equip you with the knowledge and skills to solve this equation and similar problems.

Grasping the Essence of Absolute Value: A Foundation for Success

At the heart of the equation -|-x| = -12 lies the concept of absolute value. The absolute value of a number represents its distance from zero on the number line, regardless of its sign. In simpler terms, it's the magnitude of the number without considering whether it's positive or negative. We denote the absolute value of a number x using vertical bars: |x|.

For instance, the absolute value of 7, written as |7|, is simply 7, as it is 7 units away from zero. Similarly, the absolute value of -7, written as |-7|, is also 7, because -7 is also 7 units away from zero. The absolute value function effectively transforms any negative number into its positive counterpart, while leaving positive numbers unchanged. Understanding this fundamental property is crucial for solving equations involving absolute values.

The Dance of Negative Signs: Unraveling Their Role

The equation -|-x| = -12 features two negative signs, one outside the absolute value and one on the right side of the equation. To solve this equation effectively, we need to understand how these negative signs interact with the absolute value function. The negative sign outside the absolute value acts as a multiplier, changing the sign of the entire absolute value expression. The negative sign on the right side of the equation sets the target value for the expression on the left.

Let's break it down. The absolute value |-x| will always yield a non-negative value, as we discussed earlier. However, the negative sign in front of the absolute value, -, effectively inverts this result. It transforms a non-negative value into its negative counterpart. This means that the entire expression -|-x| will always be non-positive (either negative or zero). This is a crucial observation that helps us understand the constraints on the possible values of x.

A Step-by-Step Solution: Unlocking the Solution Set of -|-x| = -12

Now, let's embark on a step-by-step journey to solve the equation -|-x| = -12. We'll break down the process into clear, manageable steps, explaining the logic behind each manipulation.

Step 1: Isolate the Absolute Value Term

The first step in our quest is to isolate the absolute value term |-x| on one side of the equation. To achieve this, we can multiply both sides of the equation by -1. This operation cancels out the negative sign in front of the absolute value, giving us:

|-x| = 12

This seemingly simple step is crucial as it simplifies the equation and allows us to focus on the absolute value term and its relationship to the constant value on the right side.

Step 2: Apply the Definition of Absolute Value: Two Paths Emerge

Now that we have isolated the absolute value term, we can invoke the definition of absolute value to understand the implications of the equation. Recall that the absolute value of a number represents its distance from zero. The equation |-x| = 12 tells us that the distance between -x and zero is 12 units.

This leads to two distinct possibilities, creating two separate paths we must explore: either -x is equal to 12 or -x is equal to -12. In other words, there are two numbers whose absolute value is 12: 12 itself and -12. This is a fundamental consequence of the absolute value function's symmetrical nature around zero.

Step 3: Solve for x: Unveiling the Individual Solutions

We now have two separate equations to solve, each representing one of the possibilities we identified in the previous step:

-x = 12

and

-x = -12

To solve for x in each case, we can multiply both sides of each equation by -1. This isolates x and reveals the individual solutions:

x = -12

and

x = 12

Step 4: Express the Solution Set: A Concise Representation

We have arrived at two potential solutions for x: -12 and 12. To express the complete set of solutions, we use set notation, which provides a concise way to represent the collection of all values that satisfy the equation. The solution set is written as:

{-12, 12}

This set contains two elements, -12 and 12, indicating that both of these values are valid solutions to the equation -|-x| = -12. No other values of x will satisfy the equation.

The Number Line Perspective: Visualizing the Solutions' Symmetry

You'll observe that both -12 and 12 are located at the same distance from zero, each being 12 units away. This visual representation beautifully illustrates the symmetrical nature of the absolute value function. The solution set {-12, 12} represents all the points on the number line that are exactly 12 units away from the origin.

Expanding Our Horizons: Tackling Similar Challenges

Now that we've successfully solved the equation -|-x| = -12, let's broaden our knowledge and tackle similar challenges. This will help you develop a more robust understanding of absolute value equations and their solutions.

Consider the equation -2|-x + 3| = -24. This equation introduces a few variations: a coefficient of -2 multiplying the absolute value term and a constant term, +3, inside the absolute value. The solution process remains similar, but we need to adapt our steps accordingly.

First, divide both sides of the equation by -2 to isolate the absolute value term: |-x + 3| = 12. Then, apply the definition of absolute value to create two equations: -x + 3 = 12 and -x + 3 = -12. Solve each equation for x to find the solution set. This example demonstrates how to handle coefficients and constants within absolute value equations.

By exploring variations like this, you'll develop a more flexible and adaptable approach to solving absolute value equations and build confidence in your problem-solving abilities.

Conclusion: Mastering the Art of Solving Absolute Value Equations

In this comprehensive exploration, we've successfully cracked the code of the equation -|-x| = -12 and determined its solution set {-12, 12}. We've delved into the fundamental concepts of absolute value, unraveled the role of negative signs, followed a methodical step-by-step solution process, and visualized the solutions on the number line. We've also expanded our horizons by tackling similar challenges.

By mastering this equation, you've gained valuable skills and insights that will empower you to tackle a wide range of mathematical problems involving absolute values. Remember the key principles: grasp the essence of absolute value, understand the dance of negative signs, and follow a methodical solution process. With practice and perseverance, you'll become a confident and proficient problem solver, capable of cracking any mathematical code that comes your way.