Solving Absolute Value Inequality |2x + 8| < 2 A Step-by-Step Guide

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This article provides a comprehensive guide to solving the absolute value inequality |2x + 8| < 2. We will break down the problem step-by-step, explaining the underlying concepts and techniques involved. By the end of this guide, you will be able to confidently solve similar absolute value inequalities and understand the reasoning behind each step. This skill is fundamental in various areas of mathematics, including algebra, calculus, and real analysis. Let's embark on this mathematical journey and demystify the process of solving absolute value inequalities.

Understanding Absolute Value

Before diving into the specifics of solving |2x + 8| < 2, it's crucial to grasp the fundamental concept of absolute value. The absolute value of a number represents its distance from zero on the number line, regardless of direction. In mathematical notation, the absolute value of a number 'x' is denoted as |x|. For example, |3| = 3 and |-3| = 3, because both 3 and -3 are 3 units away from zero. This concept is paramount when dealing with inequalities involving absolute values, as it necessitates considering both positive and negative scenarios.

The absolute value function introduces a piecewise nature to the problem. We need to consider two cases separately: the case where the expression inside the absolute value is positive or zero, and the case where it is negative. This is because the absolute value function effectively 'removes' the negative sign, and we need to account for the possibility that the expression inside the absolute value was originally negative. When dealing with absolute value inequalities, this separation into cases is a key strategy for finding the solution set. We will apply this strategy explicitly when solving |2x + 8| < 2 in the following sections. Understanding the definition and implications of absolute value is the cornerstone of solving these types of problems accurately and efficiently. Without a firm grasp of this concept, the subsequent steps might seem arbitrary or confusing. Therefore, we emphasize the importance of internalizing the idea of absolute value as a measure of distance from zero before proceeding further.

Deconstructing the Inequality |2x + 8| < 2

The absolute value inequality |2x + 8| < 2 states that the distance between the expression '2x + 8' and zero must be less than 2. This seemingly simple statement is the core of the problem we need to solve. To effectively tackle this, we need to translate this absolute value inequality into a compound inequality, which will allow us to work with standard algebraic techniques. The key principle here is recognizing that if the absolute value of an expression is less than a certain value, then the expression itself must lie between the negative and positive of that value. This is the fundamental step in converting an absolute value inequality into a solvable form.

Therefore, the inequality |2x + 8| < 2 is equivalent to the compound inequality -2 < 2x + 8 < 2. This transformation is crucial because it removes the absolute value, allowing us to treat the problem as a series of linear inequalities. This compound inequality represents two separate conditions that must be satisfied simultaneously: 2x + 8 must be greater than -2, and 2x + 8 must be less than 2. We will now proceed to solve these two inequalities individually and then find the intersection of their solutions, which will represent the solution set for the original absolute value inequality. Understanding this conversion from absolute value inequality to compound inequality is essential for solving a wide range of problems involving absolute values. It's a foundational technique that underpins many advanced mathematical concepts.

Solving the Compound Inequality

Now that we have transformed the absolute value inequality |2x + 8| < 2 into the compound inequality -2 < 2x + 8 < 2, our next step is to isolate 'x' and determine the range of values that satisfy both inequalities simultaneously. To do this, we will perform the same algebraic operations on all three parts of the inequality, maintaining the balance and ensuring the inequalities remain valid. The goal is to manipulate the inequality in such a way that 'x' is left alone in the middle, giving us a clear picture of its possible values. This process involves carefully applying the properties of inequalities and performing arithmetic operations correctly.

First, we subtract 8 from all three parts of the inequality: -2 - 8 < 2x + 8 - 8 < 2 - 8, which simplifies to -10 < 2x < -6. Next, we divide all three parts by 2: -10/2 < 2x/2 < -6/2, which simplifies to -5 < x < -3. This final inequality, -5 < x < -3, represents the solution to the original absolute value inequality. It states that 'x' must be greater than -5 and less than -3. This range of values represents all the numbers that, when substituted into the original inequality |2x + 8| < 2, will make the statement true. We have successfully isolated 'x' and found the solution set. It's important to note that each step in this process was reversible, meaning we could theoretically work backwards from the solution to the original inequality, further validating our approach.

Expressing the Solution as an Interval

The final step in solving the absolute value inequality |2x + 8| < 2 is to express the solution we found, -5 < x < -3, in interval notation. Interval notation is a concise and standard way of representing a set of numbers that lie within a specified range. It uses parentheses and brackets to indicate whether the endpoints of the interval are included in the solution set. This notation is widely used in mathematics and allows for a clear and unambiguous representation of solutions, especially when dealing with inequalities.

In this case, the inequality -5 < x < -3 indicates that 'x' can take any value strictly between -5 and -3, but it cannot be equal to -5 or -3. This is represented in interval notation using parentheses. Therefore, the solution to the inequality |2x + 8| < 2, expressed in interval notation, is (-5, -3). The parentheses signify that -5 and -3 are not included in the solution set. This notation provides a clear and compact representation of the solution, making it easy to understand and communicate the range of values that satisfy the original inequality. Understanding interval notation is crucial for accurately representing solutions to inequalities and for further work in mathematical analysis and calculus. It provides a standardized language for describing sets of numbers and their boundaries.

Conclusion

In summary, we have successfully solved the absolute value inequality |2x + 8| < 2 and expressed the solution as an interval. We began by understanding the concept of absolute value and its implications. We then transformed the absolute value inequality into a compound inequality, which allowed us to apply standard algebraic techniques to isolate 'x'. After solving the compound inequality, we arrived at the solution -5 < x < -3. Finally, we expressed this solution in interval notation as (-5, -3). This process highlights the key steps involved in solving absolute value inequalities: understanding absolute value, converting to a compound inequality, solving for the variable, and expressing the solution in the appropriate notation.

The ability to solve absolute value inequalities is a valuable skill in mathematics, with applications in various fields. By mastering the techniques outlined in this guide, you will be well-equipped to tackle similar problems and deepen your understanding of mathematical concepts. Remember to practice these steps with different examples to solidify your understanding and build confidence. The more you practice, the more fluent you will become in solving these types of problems. This skill forms a building block for more advanced topics in mathematics, making it an important area to master.

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