Solving The Equation 1/8 + 1/(x+4) = 1/x A Step-by-Step Guide
In the realm of mathematics, algebraic equations serve as the bedrock for problem-solving and analytical reasoning. These equations, characterized by variables and mathematical operations, empower us to model real-world scenarios and unravel hidden relationships. Among the diverse types of equations, rational equations, which involve fractions with variables in the denominator, often pose a unique set of challenges. In this comprehensive exploration, we delve into the intricacies of solving the rational equation 1/8 + 1/(x+4) = 1/x, meticulously dissecting each step to arrive at the solution set. Understanding how to solve such equations is crucial for various fields, including physics, engineering, and economics, where mathematical models are extensively used to describe and predict phenomena.
The journey to solving this equation will not only enhance your algebraic prowess but also deepen your understanding of mathematical principles. We will navigate through the process of eliminating fractions, simplifying the equation, and applying the quadratic formula, a powerful tool for solving second-degree polynomial equations. This exploration will underscore the importance of careful algebraic manipulation and attention to detail, essential skills for any aspiring mathematician or problem-solver. By the end of this discussion, you will be equipped with the knowledge and confidence to tackle similar rational equations, paving the way for further explorations in the captivating world of mathematics.
This detailed approach ensures clarity and comprehension, allowing you to grasp the underlying concepts and apply them effectively. So, let's embark on this mathematical journey, step by step, to unlock the solution set of this intriguing equation. Remember, every equation solved is a step forward in mastering the art of mathematical problem-solving.
The Equation Unveiled: 1/8 + 1/(x+4) = 1/x
The given equation, 1/8 + 1/(x+4) = 1/x, is a classic example of a rational equation. To embark on the journey of solving this equation, our primary objective is to eliminate the fractions. This pivotal step will transform the equation into a more manageable form, paving the way for a straightforward solution. The key to achieving this lies in identifying the least common denominator (LCD) of the fractions involved. By multiplying both sides of the equation by the LCD, we can effectively clear the denominators, thereby simplifying the equation.
The least common denominator in this case is 8x(x+4). Multiplying both sides of the equation by this LCD is a crucial step. This process ensures that each term in the equation is multiplied by the appropriate factors, effectively canceling out the denominators. As we proceed, it's essential to maintain balance and perform the same operations on both sides of the equation. This principle of equality is the cornerstone of algebraic manipulation, ensuring that the solutions remain valid.
The subsequent steps involve simplifying the resulting expression and rearranging it into a standard form, typically a quadratic equation. This form allows us to apply well-established techniques, such as factoring or the quadratic formula, to determine the values of x that satisfy the equation. Throughout this process, attention to detail is paramount. Each algebraic manipulation must be performed with precision to avoid errors that could lead to incorrect solutions. The journey from a complex rational equation to a simplified quadratic equation is a testament to the power of algebraic techniques in transforming problems into solvable forms.
By understanding the underlying principles and meticulously executing each step, we can confidently navigate the challenges posed by rational equations and arrive at the desired solutions. This methodical approach not only solves the equation at hand but also enhances our problem-solving skills, preparing us for more complex mathematical challenges in the future.
Step-by-Step Solution: A Journey Through Algebraic Manipulation
The first step in solving the equation 1/8 + 1/(x+4) = 1/x is to eliminate the fractions. To do this, we multiply both sides of the equation by the least common denominator (LCD), which is 8x(x+4). This gives us:
8x(x+4) * [1/8 + 1/(x+4)] = 8x(x+4) * [1/x]
Distributing 8x(x+4) on the left side, we get:
x(x+4) + 8x = 8(x+4)
Expanding the terms, we have:
x^2 + 4x + 8x = 8x + 32
Combining like terms, we get:
x^2 + 12x = 8x + 32
Now, we need to rearrange the equation into the standard quadratic form, which is ax^2 + bx + c = 0. Subtracting 8x and 32 from both sides, we get:
x^2 + 4x - 32 = 0
Unlocking the Quadratic: Applying the Quadratic Formula
Now that we have the quadratic equation in the standard form, x^2 + 4x - 32 = 0, we can solve for x using the quadratic formula. The quadratic formula is a powerful tool that provides the solutions for any quadratic equation of the form ax^2 + bx + c = 0. The formula is given by:
x = [-b ± √(b^2 - 4ac)] / (2a)
In our equation, a = 1, b = 4, and c = -32. Substituting these values into the quadratic formula, we get:
x = [-4 ± √(4^2 - 4 * 1 * (-32))] / (2 * 1)
Simplifying the expression under the square root:
x = [-4 ± √(16 + 128)] / 2
x = [-4 ± √144] / 2
Since √144 = 12, we have:
x = [-4 ± 12] / 2
This gives us two possible solutions for x:
x1 = (-4 + 12) / 2 = 8 / 2 = 4
x2 = (-4 - 12) / 2 = -16 / 2 = -8
Therefore, the solution set for the equation is {4, -8}.
Verifying the Solutions: Ensuring Accuracy and Validity
To ensure the accuracy and validity of our solutions, it is crucial to substitute each value of x back into the original equation and verify that the equation holds true. This process, known as verification, acts as a safeguard against potential errors made during the algebraic manipulation. By plugging the solutions back into the original equation, we can confirm whether they indeed satisfy the given conditions.
Let's first verify x = 4:
1/8 + 1/(4+4) = 1/4
1/8 + 1/8 = 1/4
2/8 = 1/4
1/4 = 1/4
This solution is valid.
Now, let's verify x = -8:
1/8 + 1/(-8+4) = 1/(-8)
1/8 + 1/(-4) = -1/8
1/8 - 2/8 = -1/8
-1/8 = -1/8
This solution is also valid.
The verification process not only confirms the correctness of our solutions but also enhances our understanding of the equation and its behavior. By observing how the solutions interact with the equation, we gain a deeper appreciation for the mathematical relationships at play. This step is an integral part of the problem-solving process, reinforcing the importance of accuracy and attention to detail in mathematical endeavors.
The Solution Set Revealed: { -8, 4 }
Having meticulously navigated the steps of solving the rational equation 1/8 + 1/(x+4) = 1/x, and after rigorously verifying our solutions, we arrive at the definitive solution set: { -8, 4 }. This set represents the values of x that satisfy the equation, making it a cornerstone of our mathematical journey.
Each element in the solution set, -8 and 4, holds significant meaning. These values are not merely numbers; they are the keys that unlock the balance within the equation. When substituted into the original equation, they create a harmonious equality, demonstrating their validity as solutions. This concept of solutions as values that satisfy an equation is fundamental to algebra and mathematical problem-solving.
Understanding the solution set is crucial for various applications. In real-world scenarios modeled by rational equations, the solutions often represent critical parameters or values. For instance, in physics, the solutions might represent the time at which a certain event occurs, or in economics, they might indicate the equilibrium price in a market. Therefore, the ability to accurately determine the solution set is not just an academic exercise; it's a practical skill with far-reaching implications.
The solution set { -8, 4 } is a testament to the power of algebraic techniques in unraveling complex problems. It encapsulates the culmination of our efforts, from eliminating fractions to applying the quadratic formula, and finally, verifying the results. This journey underscores the importance of a systematic approach, attention to detail, and a deep understanding of mathematical principles.
Conclusion: Mastering Rational Equations and Beyond
In this comprehensive exploration, we have successfully navigated the intricate path to solving the rational equation 1/8 + 1/(x+4) = 1/x. The solution set, {-8, 4}, stands as a testament to our meticulous application of algebraic techniques and our unwavering commitment to accuracy. This journey has not only provided us with the solutions to this specific equation but has also equipped us with valuable skills and insights applicable to a broader range of mathematical problems.
The process of solving rational equations, as we have seen, involves a series of strategic steps: eliminating fractions, simplifying expressions, and applying appropriate formulas such as the quadratic formula. Each step demands careful attention to detail and a solid understanding of algebraic principles. The ability to perform these steps accurately and efficiently is essential for success in mathematics and related fields.
Furthermore, the verification process underscores the importance of critical thinking and thoroughness in problem-solving. By substituting the solutions back into the original equation, we ensure their validity and reinforce our understanding of the mathematical relationships involved. This practice of verification is a hallmark of a skilled mathematician, demonstrating a commitment to accuracy and a deep appreciation for the elegance of mathematical reasoning.
As we conclude this exploration, it's important to recognize that the skills and knowledge gained extend far beyond the realm of rational equations. The principles of algebraic manipulation, problem-solving strategies, and the importance of verification are applicable to a wide array of mathematical challenges. By mastering these concepts, we empower ourselves to tackle complex problems with confidence and precision.
This journey into the world of rational equations serves as a stepping stone to further explorations in mathematics. It highlights the beauty and power of algebraic techniques in unraveling the mysteries of the mathematical universe. As we continue our mathematical journey, let us carry with us the lessons learned and the skills honed, ready to embrace new challenges and unlock new solutions.
