Solution Set Of Quadratic Inequality X² - 5 ≤ 0
In the realm of mathematics, particularly in algebra, quadratic inequalities play a crucial role. They appear frequently in various applications, ranging from physics to economics. Solving these inequalities involves finding the range of values that satisfy the given condition. This article delves into the quadratic inequality x² - 5 ≤ 0, providing a comprehensive explanation of how to determine its solution set. We will explore the underlying concepts, step-by-step methodologies, and the rationale behind each step. This exploration aims to clarify the process for students, educators, and anyone interested in enhancing their understanding of algebraic inequalities. Understanding the solution set of a quadratic inequality is fundamental in various mathematical contexts, making this a critical topic for anyone studying algebra and beyond. The solution set is the set of all values of x that make the inequality true. In the case of x² - 5 ≤ 0, we are looking for all real numbers x such that when x is squared and then 5 is subtracted, the result is either negative or zero.
The core question we address in this article is: What is the solution set of the quadratic inequality x² - 5 ≤ 0? To answer this, we must methodically solve the inequality. This involves a few key steps, which we will detail in the subsequent sections. The provided options are:
A. {x | -5 ≤ x ≤ 5} B. {x | -√5 ≤ x ≤ 5} C. {x | -5 ≤ x ≤ √5} D. {x | -√5 ≤ x ≤ √5}
Each of these options represents a range of values for x, and only one of them correctly describes the solution set for the inequality. Our goal is to determine which of these options is the correct one. This requires a clear understanding of how to manipulate the inequality algebraically and interpret the result. We will walk through the process, ensuring that each step is well-explained and justified. This methodical approach will not only help in solving this particular problem but also in tackling similar inequalities in the future. By the end of this article, readers should be confident in their ability to identify the solution sets of quadratic inequalities.
Step 1: Rearrange the Inequality
The initial step in solving the quadratic inequality x² - 5 ≤ 0 is to isolate the x² term. This is achieved by adding 5 to both sides of the inequality. Doing so gives us x² ≤ 5. This step is crucial as it simplifies the inequality into a form that is easier to work with. By isolating the x² term, we can directly address the squared variable and its relationship to the constant value on the other side of the inequality. This rearrangement is a standard technique in solving inequalities and equations, and it helps to clearly define the conditions that x must satisfy. The inequality x² ≤ 5 now provides a clearer picture of the possible values of x that will make the statement true. It tells us that the square of x must be less than or equal to 5. This is a significant simplification that sets the stage for the next steps in finding the solution set.
Step 2: Take the Square Root
The next step involves taking the square root of both sides of the inequality x² ≤ 5. When taking the square root of an inequality, it is essential to remember that we must consider both the positive and negative square roots. This is because squaring either a positive or a negative number will result in a positive number. Therefore, we obtain two inequalities: -√5 ≤ x and x ≤ √5. This step is critical because it directly addresses the square root operation and its implications for the solution set. By considering both positive and negative roots, we ensure that we capture all possible values of x that satisfy the original inequality. The two resulting inequalities define the bounds within which x must lie. The first inequality, -√5 ≤ x, tells us that x must be greater than or equal to the negative square root of 5. The second inequality, x ≤ √5, tells us that x must be less than or equal to the positive square root of 5. These two conditions together define the range of values for x that will satisfy the original inequality.
Step 3: Express the Solution Set
Combining the two inequalities, -√5 ≤ x and x ≤ √5, we can express the solution set as a single compound inequality: -√5 ≤ x ≤ √5. This compound inequality states that x must be greater than or equal to -√5 and less than or equal to √5. In set notation, this is represented as {x | -√5 ≤ x ≤ √5}. This step is the culmination of the previous steps, bringing together the two individual inequalities into a cohesive expression that defines the solution set. The compound inequality is a concise and clear way to represent the range of values that x can take while still satisfying the original inequality. The set notation provides an alternative way to express the same solution, using mathematical symbols to precisely define the set of all x values that are part of the solution. This final form of the solution set is what we will compare to the given options to determine the correct answer.
Now, let's analyze the given options and compare them with our derived solution set:
A. {x | -5 ≤ x ≤ 5} B. {x | -√5 ≤ x ≤ 5} C. {x | -5 ≤ x ≤ √5} D. {x | -√5 ≤ x ≤ √5}
Comparing these options with our solution set, {x | -√5 ≤ x ≤ √5}, it becomes clear that option D matches our result. Options A, B, and C include ranges that are either too broad or incorrect based on our calculations. This step is crucial for verifying the correctness of our solution. By carefully comparing each option with the derived solution set, we can identify any discrepancies and ensure that we have arrived at the correct answer. Option A, for example, includes values outside the range defined by -√5 and √5. Options B and C have one bound that is correct but the other bound is incorrect, indicating a misunderstanding of the inequality's limits. Only option D perfectly aligns with the solution set we obtained through our step-by-step process, confirming its validity.
Therefore, the correct solution set for the quadratic inequality x² - 5 ≤ 0 is D. {x | -√5 ≤ x ≤ √5}. This comprehensive explanation demonstrates the process of solving quadratic inequalities, emphasizing the importance of considering both positive and negative roots when dealing with square roots in inequalities. The step-by-step methodology ensures clarity and accuracy, making it easier to understand the underlying principles. This conclusion solidifies our understanding of the problem and its solution. By correctly identifying the solution set, we have demonstrated our ability to manipulate and solve quadratic inequalities. The detailed explanation provided throughout this article serves as a valuable resource for anyone looking to improve their skills in algebra and mathematical problem-solving. Understanding how to solve quadratic inequalities is not only essential for academic success but also for various real-world applications where mathematical models are used to represent and solve problems. The techniques and principles discussed here can be applied to a wide range of similar problems, making this a fundamental skill in mathematics.
The final answer is (D).
