Solving 1/(6x²) = 1/(3x²) - 1/x A Comprehensive Guide
In this article, we will delve into the process of solving rational equations, focusing specifically on the equation 1/(6x²) = 1/(3x²) - 1/x. Rational equations, which involve fractions with polynomials in the numerator and denominator, are a fundamental topic in algebra. Mastering the techniques to solve them is crucial for success in higher-level mathematics and various applications in science and engineering. This comprehensive guide will walk you through the steps to solve the given equation, while also providing insights into the underlying principles and potential pitfalls. Solving rational equations requires a systematic approach, and our goal is to equip you with the knowledge and skills to tackle such problems confidently. Understanding the nuances of rational expressions and equations is key to simplifying complex problems into manageable steps. By the end of this article, you will have a clear understanding of how to solve 1/(6x²) = 1/(3x²) - 1/x and similar equations.
Before we dive into the specifics of solving rational equations, it’s important to understand what they are and the key principles involved. A rational equation is simply an equation that contains one or more rational expressions. A rational expression, in turn, is a fraction where the numerator and/or the denominator are polynomials. Examples of rational expressions include 1/x, (x+1)/(x-2), and (3x² + 2x - 1)/(x² + 4). The presence of variables in the denominator is what distinguishes rational equations from simpler algebraic equations. This characteristic also introduces some important considerations, such as the possibility of extraneous solutions. When dealing with rational equations, it's crucial to identify values of the variable that would make the denominator zero, as division by zero is undefined. These values are called restrictions and must be excluded from the solution set. The process of solving rational equations typically involves eliminating the fractions by multiplying both sides of the equation by a common denominator. This step transforms the rational equation into a polynomial equation, which can then be solved using standard algebraic techniques. However, it's essential to check the solutions obtained against the original equation to ensure they are valid and not extraneous. This verification step is a critical part of the process and helps to avoid errors. In the following sections, we will apply these principles to solve the equation 1/(6x²) = 1/(3x²) - 1/x, providing a step-by-step explanation of each stage. By understanding the fundamental concepts, you will be well-prepared to tackle a wide range of rational equations. Understanding these principles sets the stage for effectively solving the equation at hand.
To solve the rational equation 1/(6x²) = 1/(3x²) - 1/x, we will follow a step-by-step approach to ensure clarity and accuracy. First, we need to identify any restrictions on the variable x. Since the denominators in the equation are 6x², 3x², and x, the variable x cannot be equal to zero. This is because division by zero is undefined, and any solution that results in a zero denominator must be excluded. So, x ≠ 0 is our primary restriction. Next, we want to eliminate the fractions to simplify the equation. To do this, we find the least common denominator (LCD) of the fractions. The denominators are 6x², 3x², and x. The LCD is 6x². Multiplying both sides of the equation by the LCD, 6x², we get:
6x² * [1/(6x²)] = 6x² * [1/(3x²) - 1/x]
This simplifies to:
1 = 6x² * (1/(3x²)) - 6x² * (1/x)
Further simplification yields:
1 = 2 - 6x
Now, we have a linear equation that we can easily solve. Subtracting 2 from both sides, we get:
-1 = -6x
Dividing both sides by -6, we find:
x = 1/6
Finally, we need to check if this solution is valid by ensuring it does not violate our restriction x ≠ 0. Since 1/6 is not equal to 0, it is a valid solution. Therefore, the solution to the equation 1/(6x²) = 1/(3x²) - 1/x is x = 1/6. This step-by-step approach illustrates how to systematically solve rational equations, emphasizing the importance of identifying restrictions and verifying solutions. With this method, you can confidently tackle similar problems.
After finding a potential solution to a rational equation, it's absolutely critical to verify that the solution is valid. This verification step is necessary because multiplying both sides of an equation by an expression that contains a variable can sometimes introduce extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but do not satisfy the original equation. In our case, we found the solution x = 1/6 for the equation 1/(6x²) = 1/(3x²) - 1/x. To verify this solution, we will substitute x = 1/6 back into the original equation and check if both sides of the equation are equal. Substituting x = 1/6 into the left side of the equation, we get:
1/(6*(1/6)²) = 1/(6*(1/36)) = 1/(1/6) = 6
Now, we substitute x = 1/6 into the right side of the equation:
1/(3*(1/6)²) - 1/(1/6) = 1/(3*(1/36)) - 6 = 1/(1/12) - 6 = 12 - 6 = 6
Since both sides of the equation are equal to 6 when x = 1/6, the solution is valid. This verification step confirms that x = 1/6 is indeed the correct solution to the rational equation 1/(6x²) = 1/(3x²) - 1/x. By performing this check, we ensure that our solution is not an extraneous solution and that it satisfies the original equation. This meticulous approach is a hallmark of problem-solving in mathematics and is crucial for obtaining accurate results. Always take the time to verify your solutions, especially when dealing with rational equations and other types of equations that can produce extraneous solutions. The process of verifying the solution reinforces the correctness of our answer and confirms our understanding of the problem.
When solving rational equations, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and improve your problem-solving accuracy. One of the most frequent mistakes is failing to identify and exclude restrictions on the variable. As we discussed earlier, any value of the variable that makes the denominator of a fraction equal to zero must be excluded from the solution set. For example, in the equation 1/(6x²) = 1/(3x²) - 1/x, we identified that x ≠ 0. Forgetting this restriction can lead to incorrect solutions. Another common mistake is incorrectly finding the least common denominator (LCD). The LCD must be a multiple of all the denominators in the equation. Errors in determining the LCD can result in an incorrect simplification of the equation. Always double-check your LCD to ensure it is the smallest expression that is divisible by all denominators. A third mistake is not distributing properly when multiplying both sides of the equation by the LCD. It's crucial to multiply every term on both sides of the equation by the LCD to maintain equality. Failing to distribute correctly can lead to an unbalanced equation and an incorrect solution. Perhaps the most critical mistake is forgetting to verify the solutions. As we emphasized in the previous section, extraneous solutions can arise when solving rational equations. Always substitute your potential solutions back into the original equation to check for validity. This step is non-negotiable and can save you from accepting an incorrect answer. By being mindful of these common mistakes, you can approach rational equations with greater confidence and accuracy. Practicing these strategies will significantly improve your ability to solve these types of problems effectively. Avoiding these errors is essential for success in algebra and beyond.
While we have demonstrated a standard method for solving the rational equation 1/(6x²) = 1/(3x²) - 1/x, it's beneficial to explore alternative approaches. Different methods can provide additional insights and may be more efficient in certain situations. One alternative method involves combining the fractions on one side of the equation before eliminating the denominators. Starting with the original equation:
1/(6x²) = 1/(3x²) - 1/x
We can subtract 1/(3x²) from both sides to get:
1/(6x²) - 1/(3x²) = -1/x
Now, we find a common denominator for the left side, which is 6x²:
(1 - 2)/(6x²) = -1/x
This simplifies to:
-1/(6x²) = -1/x
To eliminate the fractions, we can cross-multiply:
-1 * x = -1 * 6x²
Which gives us:
-x = -6x²
Now, we move all terms to one side:
6x² - x = 0
We can factor out an x:
x(6x - 1) = 0
This gives us two potential solutions: x = 0 and 6x - 1 = 0. Solving 6x - 1 = 0, we get x = 1/6. However, we must remember our restriction that x ≠ 0. Therefore, x = 0 is an extraneous solution, and the only valid solution is x = 1/6. This alternative method, while slightly different in its initial steps, arrives at the same solution as our previous method. It demonstrates that there can be multiple pathways to solving a rational equation, and choosing the most efficient method often depends on the specific equation and your personal preference. Exploring different approaches enhances your problem-solving skills and provides a deeper understanding of the underlying concepts. These alternative methods underscore the flexibility in mathematical problem-solving.
Rational equations are not just abstract mathematical concepts; they have significant real-world applications in various fields. Understanding these applications can help you appreciate the practical relevance of solving equations like 1/(6x²) = 1/(3x²) - 1/x. In physics, rational equations are used to describe relationships between quantities such as distance, rate, and time. For example, if you have two objects moving at different speeds covering the same distance, you might use a rational equation to find the time it takes for each object to complete the journey. The equation might involve terms representing the speeds and the total distance. In engineering, rational equations are crucial for designing circuits and analyzing electrical systems. The relationships between voltage, current, and resistance in a circuit can often be modeled using rational equations. Engineers use these equations to ensure that circuits function correctly and to optimize their performance. In chemistry, rational equations can be used to model reaction rates and equilibrium conditions. The concentrations of reactants and products in a chemical reaction are often related through rational expressions, and solving the resulting equations helps chemists understand how reactions proceed and reach equilibrium. In economics, rational equations can appear in models of supply and demand. For instance, the price elasticity of demand, which measures how the quantity demanded of a good responds to a change in its price, can be expressed as a rational function. Economists use these equations to analyze market behavior and make predictions about future prices and quantities. These are just a few examples of the many real-world applications of rational equations. By mastering the techniques to solve these equations, you are developing skills that are valuable in a wide range of disciplines. The ability to apply mathematical concepts to practical problems is a key aspect of both academic and professional success. These applications illustrate the broad utility of the concepts discussed.
In this comprehensive guide, we have explored the process of solving the rational equation 1/(6x²) = 1/(3x²) - 1/x. We began by understanding the basics of rational equations and the importance of identifying restrictions on the variable. We then walked through a step-by-step solution, highlighting the need to eliminate fractions by multiplying by the least common denominator and the necessity of verifying the solution to avoid extraneous results. We found that the solution to the equation is x = 1/6. Furthermore, we discussed common mistakes to avoid when solving rational equations, such as failing to identify restrictions, incorrectly finding the LCD, and forgetting to verify solutions. By being aware of these pitfalls, you can improve your accuracy and confidence in solving similar problems. We also explored an alternative method for solving the equation, demonstrating that there can be multiple approaches to the same problem. This flexibility in problem-solving is a valuable skill in mathematics and beyond. Finally, we discussed real-world applications of rational equations in fields such as physics, engineering, chemistry, and economics, illustrating the practical relevance of these concepts. By mastering the techniques presented in this guide, you will be well-equipped to tackle a variety of rational equations and apply them in different contexts. The skills you have gained are not only valuable for academic pursuits but also for real-world problem-solving. We hope this guide has provided you with a solid understanding of how to solve rational equations and has inspired you to continue exploring the fascinating world of mathematics. Understanding these concepts is crucial for continued success in mathematics and related fields.
