Solving The Inequality |x-4| < 3 A Step-by-Step Guide
When faced with absolute value inequalities, such as the one presented, |x-4| < 3, it's crucial to understand the underlying concept of absolute value. The absolute value of a number represents its distance from zero on the number line. Therefore, the inequality |x-4| < 3 signifies that the distance between x and 4 is less than 3. This understanding forms the foundation for solving such inequalities effectively.
To effectively address absolute value inequalities, we must decompose them into two separate cases, each representing a different scenario within the absolute value's definition. This approach is essential for capturing the full range of solutions that satisfy the inequality. In the given inequality, |x-4| < 3, the expression inside the absolute value, x-4, can either be positive or negative. Recognizing these two possibilities is the key to unlocking the solution. By meticulously examining each case, we can accurately determine the values of x that meet the condition set forth by the inequality.
Let's delve into the first case, where the expression x-4 is considered positive or zero. In this scenario, the absolute value bars simply vanish, and the inequality transforms into x-4 < 3. This simplification allows us to tackle the inequality using standard algebraic techniques, providing a direct path towards isolating x and identifying a portion of the solution set. By focusing on this positive case, we gain valuable insights into the behavior of x within a specific range, contributing to our overall understanding of the solution.
Now, let's transition to the second case, where the expression x-4 is negative. When dealing with a negative expression inside the absolute value, we introduce a crucial step: multiplying the expression by -1 before removing the absolute value bars. This step ensures that we are working with the positive equivalent of the distance from zero. Consequently, our inequality transforms into -(x-4) < 3, which can be further simplified to -x + 4 < 3. This transformation is vital for accurately representing the condition when x-4 is negative, paving the way for solving the inequality in this specific context. By addressing both the positive and negative cases, we ensure a comprehensive and accurate solution to the absolute value inequality.
By considering both scenarios, we create two distinct inequalities that must be solved independently. These inequalities will then provide us with the range of values for x that satisfy the original absolute value inequality. This method ensures that we capture all possible solutions, leaving no room for error. The meticulous approach of dissecting the problem into manageable cases is a hallmark of effective problem-solving in mathematics.
In our first case, we assume that the expression inside the absolute value, x - 4, is non-negative. This allows us to directly remove the absolute value signs, resulting in the straightforward inequality x - 4 < 3. Solving this inequality involves a single, elegant step: adding 4 to both sides. This operation isolates x on the left side, providing us with a clear upper bound for its value. The resulting inequality, x < 7, reveals that all values of x less than 7 satisfy the original inequality under the condition that x - 4 is non-negative. This is a crucial piece of the puzzle, defining one portion of the solution set.
The simplicity of this step belies its importance. By adding 4 to both sides, we perform a fundamental algebraic manipulation that preserves the inequality while simultaneously bringing us closer to the solution. This highlights the power of basic algebraic principles in unraveling more complex problems. The clarity of the result, x < 7, underscores the directness and efficiency of this approach. It's a testament to the elegance of mathematical problem-solving when applied judiciously.
In the second case, we address the scenario where the expression inside the absolute value, x - 4, is negative. This requires us to take an additional step before removing the absolute value signs. We multiply the expression by -1, effectively transforming it into its positive counterpart. This is crucial because the absolute value represents the distance from zero, which is always a non-negative quantity. Applying this transformation to our inequality, we get -(x - 4) < 3, which can be further simplified to -x + 4 < 3.
Now, to solve this inequality, we embark on a series of algebraic manipulations aimed at isolating x. First, we subtract 4 from both sides of the inequality, yielding -x < -1. This step isolates the term containing x on the left side, paving the way for the final step. However, we must be cautious when dealing with a negative coefficient in front of x. To obtain a positive x, we multiply both sides of the inequality by -1. It's paramount to remember that multiplying or dividing an inequality by a negative number reverses the direction of the inequality sign. Thus, -x < -1 transforms into x > 1.
This seemingly simple transformation is a critical element in solving inequalities. The act of reversing the inequality sign is not merely a mechanical step; it's a reflection of the fundamental properties of inequalities. Multiplying by a negative number flips the order of the number line, and the inequality sign must be adjusted accordingly. The result, x > 1, reveals that all values of x greater than 1 satisfy the original inequality under the condition that x - 4 is negative. This complements the solution obtained in the first case, providing a complete picture of the solution set.
Having meticulously solved both cases, we now possess two crucial pieces of information: x < 7 and x > 1. These inequalities represent the conditions under which the original absolute value inequality, |x-4| < 3, holds true. However, our task is not yet complete. We must synthesize these individual solutions into a cohesive and comprehensive answer that accurately captures the entire solution set. This involves recognizing that x must simultaneously satisfy both conditions. In essence, we are seeking the intersection of the two solution sets.
To achieve this synthesis, we can express the combined solution in a variety of ways, each offering a slightly different perspective on the same underlying reality. One particularly insightful approach is to use a compound inequality. This notation allows us to elegantly represent the range of values that x can take while satisfying both conditions. The compound inequality directly expresses the bounds within which x must reside, providing a clear and concise representation of the solution set.
Alternatively, we can leverage interval notation, a powerful tool for representing sets of real numbers. Interval notation employs parentheses and brackets to denote whether the endpoints of an interval are included or excluded. This notation is particularly well-suited for expressing continuous ranges of values, such as those encountered in inequalities. By using interval notation, we can succinctly capture the solution set in a manner that is both mathematically rigorous and easily interpretable.
Furthermore, a graphical representation can provide valuable intuition and visual confirmation of our solution. By plotting the inequalities on a number line, we can visually identify the region where the solutions overlap. This graphical approach not only reinforces our understanding but also serves as a powerful check against potential errors. The number line provides a clear and immediate depiction of the solution set, making it readily accessible to both novice and expert problem solvers.
Integrating the solutions from both cases, x < 7 and x > 1, we arrive at the complete solution to the inequality |x-4| < 3. This solution can be expressed elegantly using a compound inequality: 1 < x < 7. This notation succinctly conveys that x must be greater than 1 and simultaneously less than 7. It precisely defines the range of values that satisfy the original inequality, leaving no room for ambiguity.
Alternatively, we can represent this solution using interval notation. In this notation, the solution set is written as (1, 7). The parentheses indicate that the endpoints, 1 and 7, are not included in the solution set. This notation provides a compact and mathematically rigorous way to express the continuous range of values that x can take. Interval notation is a valuable tool for mathematicians and scientists, allowing for the precise communication of solution sets.
To further solidify our understanding, let's consider the implications of this solution. It tells us that any number between 1 and 7, excluding 1 and 7 themselves, will satisfy the inequality |x-4| < 3. For example, values like 2, 4, and 6 fall within this range and will indeed make the inequality true. Conversely, values outside this range, such as 0, 1, 7, and 8, will not satisfy the inequality. This concrete interpretation helps to bridge the gap between the abstract mathematical solution and its practical meaning.
In the context of the original multiple-choice question, we can now definitively identify the correct answer. The solution 1 < x < 7 corresponds to option B. 1 within the given choices. This underscores the importance of a thorough and systematic approach to problem-solving. By carefully considering both cases of the absolute value inequality, we have arrived at a precise and accurate solution. This methodical approach not only provides the correct answer but also enhances our understanding of the underlying mathematical principles.
To fully appreciate the correctness of our solution, it's instructive to examine why the other options presented are incorrect. This process not only reinforces our understanding of the solution but also hones our critical thinking skills in mathematics. By identifying the flaws in alternative approaches, we solidify our grasp of the correct method and develop a deeper appreciation for the nuances of mathematical problem-solving.
Option A. -7 presents a single numerical value. While -7 might seem related to the problem due to the presence of the number 7 in the inequality, it fails to capture the entire range of solutions. The inequality |x-4| < 3 defines a continuous interval of values, not just a single point. Therefore, -7, by itself, is an insufficient answer.
Option C. x < -7 or x < -1 suggests a solution set that extends to negative infinity. However, this contradicts the constraints imposed by the absolute value. The absolute value ensures that the distance between x and 4 is less than 3, which inherently limits the range of possible values for x. The inequalities in option C do not reflect this limitation, making them an incorrect representation of the solution set.
Option D. x > 1 or x < 7 presents two separate inequalities. While each inequality individually captures a portion of the solution, the use of "or" implies that x needs to satisfy only one of the conditions. This is misleading because x must simultaneously satisfy both conditions to fall within the solution set of the original absolute value inequality. The "or" connective leads to an overinclusive solution, encompassing values that do not actually satisfy the original inequality.
In conclusion, solving the absolute value inequality |x-4| < 3 requires a careful consideration of both positive and negative cases. By breaking down the problem into manageable steps and applying fundamental algebraic principles, we arrive at the solution 1 < x < 7, which can also be expressed in interval notation as (1, 7). This solution represents the range of values for x that satisfy the given inequality. Understanding the properties of absolute values and inequalities is crucial for accurately solving such problems and avoiding common pitfalls. The correct answer among the given options is B. 1 < x < 7. Understanding absolute value inequalities is not just about finding the right answer; it's about developing a deeper appreciation for mathematical reasoning and problem-solving strategies. By mastering these concepts, you'll be well-equipped to tackle a wide range of mathematical challenges.
