Solving 2|x+5.3|=4.2 Absolute Value Equation Step By Step

by ADMIN 58 views

In the realm of mathematics, solving equations is a fundamental skill. Among the various types of equations, absolute value equations often pose a unique challenge. These equations involve the absolute value of an expression, which represents its distance from zero on the number line. This article delves into the process of solving the absolute value equation 2|x+5.3|=4.2, providing a step-by-step guide to finding the values of x that satisfy the equation. Understanding how to solve absolute value equations is crucial for success in algebra and beyond. The ability to manipulate these equations and arrive at the correct solutions demonstrates a strong grasp of mathematical principles and problem-solving techniques. This article aims to break down the complexities of absolute value equations, making the process accessible and understandable for students and enthusiasts alike. By exploring the intricacies of this equation, we not only find the specific solutions but also deepen our understanding of the broader concepts that govern absolute value functions and their applications in various mathematical contexts.

Before diving into the solution, it's essential to grasp the concept of absolute value. The absolute value of a number is its distance from zero, regardless of direction. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5. This is denoted by vertical bars: |5| = 5 and |-5| = 5. Understanding this concept is crucial when dealing with absolute value equations. Absolute value equations often have two possible solutions because the expression inside the absolute value bars can be either positive or negative, but the result will always be a positive distance from zero. This duality is a key aspect to consider when setting up and solving these equations. Recognizing that both positive and negative values within the absolute value can lead to the same result is the first step in tackling these problems effectively. By keeping this principle in mind, we can approach absolute value equations with a clear understanding of the potential outcomes and how to find them. This foundational knowledge is not only essential for solving this specific equation but also for handling a wide range of absolute value problems in the future.

Now, let's break down the given equation: 2|x+5.3|=4.2. Our goal is to isolate x, but we need to address the absolute value first. The equation essentially states that twice the absolute value of (x+5.3) equals 4.2. This means that the expression inside the absolute value bars, (x+5.3), can be either a positive or negative value that, when multiplied by 2 after taking the absolute value, results in 4.2. To solve this, we need to consider two separate cases, one where (x+5.3) is positive and one where it is negative. This approach is fundamental to solving absolute value equations, as it accounts for the dual nature of absolute value. By carefully considering both scenarios, we ensure that we capture all possible solutions for x. This systematic approach not only helps in solving this particular equation but also provides a framework for tackling other absolute value problems. The initial step of isolating the absolute value term is crucial, as it sets the stage for the subsequent steps of splitting the equation into two cases and solving each one independently.

Step 1: Isolate the Absolute Value

To begin, we need to isolate the absolute value term. Divide both sides of the equation by 2:

|x+5.3| = 4.2 / 2

|x+5.3| = 2.1

This step simplifies the equation and makes it easier to work with. By isolating the absolute value, we are essentially setting the stage for the next crucial step: considering the two possible scenarios that arise from the definition of absolute value. This isolation is a standard technique in solving absolute value equations and is crucial for accurately determining the solutions. The simplification achieved in this step makes the subsequent algebraic manipulations more straightforward and less prone to error. By focusing on isolating the absolute value, we can clearly see the core relationship we need to solve: the distance of (x+5.3) from zero is 2.1. This clear understanding is pivotal for proceeding with the solution.

Step 2: Consider Both Positive and Negative Cases

Now, we consider two cases:

Case 1: x + 5.3 = 2.1

Case 2: x + 5.3 = -2.1

This is because the expression inside the absolute value can be either 2.1 or -2.1, as both have an absolute value of 2.1. This step is the heart of solving absolute value equations. It directly addresses the dual nature of absolute value, where both positive and negative values within the absolute value bars can result in the same positive value. By splitting the equation into these two cases, we ensure that we don't miss any potential solutions. Each case represents a separate linear equation that can be solved independently. The careful consideration of both cases is what distinguishes solving absolute value equations from solving regular algebraic equations. This step highlights the importance of understanding the underlying principles of absolute value and applying them systematically to find all possible solutions.

Step 3: Solve for x in Each Case

Let's solve for x in each case:

Case 1:

x + 5.3 = 2.1

x = 2.1 - 5.3

x = -3.2

Case 2:

x + 5.3 = -2.1

x = -2.1 - 5.3

x = -7.4

These calculations give us two possible values for x. Solving for x in each case involves basic algebraic manipulation, specifically subtracting 5.3 from both sides of the equation. This step is a straightforward application of algebraic principles to isolate the variable. The resulting values, -3.2 and -7.4, are the potential solutions to the original absolute value equation. Each solution corresponds to one of the two cases we considered, highlighting the importance of addressing both possibilities when dealing with absolute value. These solutions indicate the specific values of x that make the expression inside the absolute value bars equal to either 2.1 or -2.1, thus satisfying the original equation. The accuracy of these calculations is crucial for arriving at the correct final answer.

Therefore, the values of x that satisfy the equation are -3.2 and -7.4.

In conclusion, the correct answer is C: x can equal -3.2 or -7.4. This solution demonstrates the importance of understanding the properties of absolute value and applying them methodically to solve equations. The process involved isolating the absolute value, considering both positive and negative cases, and solving for x in each case. This approach is applicable to a wide range of absolute value problems and underscores the significance of a systematic approach to mathematical problem-solving. The final answer reinforces the concept that absolute value equations often have two solutions due to the nature of absolute value as a measure of distance from zero. The ability to accurately solve such equations is a valuable skill in mathematics and related fields.

Now, let's map our solutions back to the given options:

A. x can only equal -3.2. (Incorrect)

B. x can only equal 7.4. (Incorrect)

C. x can equal -3.2 or -7.4. (Correct)

D. x can equal -3.2 or 7.4. (Incorrect)

Only option C accurately reflects the solutions we found. This step is crucial in any problem-solving process, especially in multiple-choice scenarios. It involves comparing the calculated solutions with the available options and selecting the one that matches perfectly. In this case, we found two solutions, -3.2 and -7.4, and option C is the only one that includes both these values. This final check ensures that we have not made any errors in our calculations or interpretations. It also reinforces the importance of carefully reading and understanding the question and the answer choices. The ability to correctly identify the matching option demonstrates a thorough understanding of the problem and the solution process.

Mastering absolute value equations is a crucial skill in algebra and beyond. This article has provided a detailed walkthrough of solving the equation 2|x+5.3|=4.2, highlighting the key steps involved: isolating the absolute value, considering both positive and negative cases, and solving for x in each case. By understanding these steps, students can confidently tackle a variety of absolute value problems. The ability to solve these equations not only strengthens algebraic skills but also enhances problem-solving abilities in general. The systematic approach outlined in this article can be applied to other mathematical problems, emphasizing the importance of a structured and logical approach. Furthermore, understanding absolute value is fundamental for more advanced mathematical concepts, such as inequalities and functions. Therefore, mastering this topic is an investment in future mathematical success. By consistently practicing and applying these techniques, students can build a strong foundation in algebra and develop the confidence to tackle more complex mathematical challenges.

  • Absolute Value Equations
  • Solving Equations
  • Algebra
  • Mathematics
  • Problem-Solving
  • Equation Solutions
  • Isolating Variables
  • Positive and Negative Cases
  • Step-by-Step Guide
  • Math Education