Solving Rational Equations: A Step-by-Step Guide To 2/(x+8) = 4/x

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Understanding Rational Equations

In the realm of mathematics, solving equations is a fundamental skill. Among the various types of equations, rational equations hold a significant place. Rational equations are equations that contain fractions with variables in the denominator. They often appear in various real-world applications, such as physics, engineering, and economics. This article dives deep into the process of solving a specific rational equation: 2/(x+8) = 4/x. We will explore the steps involved, the underlying principles, and potential pitfalls to avoid. Before we jump into the solution, let's clarify what a rational equation is and why solving them requires a specific approach. A rational equation, at its core, is an equation where one or both sides are rational expressions. A rational expression is simply a fraction where the numerator and/or the denominator are polynomials. Because variables appear in the denominators, we need to be mindful of values that could make the denominator zero, as division by zero is undefined. These values are called extraneous solutions and must be identified and excluded from the final answer.

The importance of understanding rational equations extends beyond academic exercises. Many practical problems can be modeled using these equations. For instance, in physics, rational equations can describe the relationship between resistance, voltage, and current in electrical circuits. In engineering, they can be used to model fluid flow or heat transfer. In economics, rational equations can help analyze supply and demand curves. Therefore, mastering the techniques to solve these equations equips you with valuable tools for various applications. As we proceed, we will break down the process into manageable steps, ensuring clarity and understanding. From finding the common denominator to checking for extraneous solutions, each step will be explained in detail. Let's begin by outlining the specific equation we will be tackling: 2/(x+8) = 4/x. This equation serves as an excellent example to illustrate the general strategy for solving rational equations. Our goal is to isolate the variable 'x' and determine its value(s) that satisfy the equation. By the end of this article, you will not only be able to solve this particular equation but also have a solid foundation for tackling a wide range of rational equations.

Step 1: Identifying the Domain

Domain restrictions are a crucial aspect of rational equations, and understanding them is paramount before diving into the solution. As mentioned earlier, rational equations involve fractions with variables in the denominator. The golden rule of mathematics is that division by zero is undefined. Therefore, we must identify any values of the variable that would make the denominator zero and exclude them from our possible solutions. This set of permissible values is called the domain of the equation. Let's apply this concept to our equation: 2/(x+8) = 4/x. We have two denominators: (x+8) and x. To find the values that make these denominators zero, we set each one equal to zero and solve for x.

For the first denominator, (x+8), we have the equation x+8 = 0. Subtracting 8 from both sides, we get x = -8. This means that if x is -8, the denominator (x+8) becomes zero, and the fraction 2/(x+8) is undefined. Therefore, x = -8 is a restricted value and cannot be a solution to our equation. Similarly, for the second denominator, x, we have the simple equation x = 0. This means that if x is 0, the denominator x becomes zero, and the fraction 4/x is undefined. Thus, x = 0 is another restricted value. Now that we have identified the restricted values, we can formally define the domain of our equation. The domain is the set of all real numbers except for -8 and 0. We can express this mathematically as: x ≠ -8 and x ≠ 0. These restrictions are essential because any solution we find must be within this domain. If we arrive at a solution that is either -8 or 0, we must discard it as an extraneous solution. Failing to consider the domain can lead to incorrect answers. In the subsequent steps, we will proceed with the algebraic manipulations to solve the equation, but we will always keep in mind the restrictions on x. This vigilance will ensure that our final solution is valid and meaningful. Understanding domain restrictions is not just a technicality; it's a fundamental part of solving rational equations and ensuring the mathematical integrity of our results. In the next step, we will explore how to clear the fractions from the equation, making it easier to solve.

Step 2: Clearing the Fractions

The presence of fractions in an equation can often make it appear more daunting. However, a powerful technique called "clearing the fractions" simplifies the equation and transforms it into a more manageable form. This technique involves multiplying both sides of the equation by the least common denominator (LCD) of all the fractions present. The LCD is the smallest expression that is divisible by all the denominators in the equation. In our equation, 2/(x+8) = 4/x, we have two denominators: (x+8) and x. To find the LCD, we need to identify the factors present in each denominator. In this case, the denominators are already in their simplest factored form. The LCD is simply the product of these distinct factors: x(x+8). Now that we have the LCD, we can proceed to clear the fractions. We multiply both sides of the equation by x(x+8): x(x+8) * [2/(x+8)] = x(x+8) * [4/x] The key to this step is to distribute the LCD correctly on both sides. On the left side, the (x+8) terms cancel out, leaving us with: 2x. On the right side, the x terms cancel out, leaving us with: 4(x+8). Our equation now looks much simpler: 2x = 4(x+8). We have successfully eliminated the fractions, transforming the rational equation into a linear equation. This is a significant step forward because linear equations are generally easier to solve. However, it's crucial to remember that multiplying both sides of an equation by an expression containing a variable can sometimes introduce extraneous solutions. This is why it's essential to check our solutions against the original equation's domain, as we discussed in Step 1. Before we move on, let's recap what we've accomplished. We identified the LCD as x(x+8) and multiplied both sides of the equation by it. This cleared the fractions, resulting in the simpler equation 2x = 4(x+8). In the next step, we will focus on solving this linear equation for x. We will distribute, combine like terms, and isolate the variable to find its value. Clearing the fractions is a fundamental technique in solving rational equations, and mastering it is essential for success.

Step 3: Solving the Simplified Equation

Having successfully cleared the fractions in the previous step, we are now left with a simpler equation: 2x = 4(x+8). This is a linear equation, which can be solved using standard algebraic techniques. Our goal is to isolate the variable 'x' on one side of the equation. The first step in solving this equation is to distribute the 4 on the right side: 2x = 4x + 32. Next, we want to get all the 'x' terms on one side of the equation. We can subtract 4x from both sides: 2x - 4x = 4x + 32 - 4x. This simplifies to: -2x = 32. Now, to isolate 'x', we divide both sides by -2: (-2x) / -2 = 32 / -2. This gives us: x = -16. We have found a potential solution for x: -16. However, it is crucial to remember the domain restrictions we identified in Step 1. We determined that x cannot be -8 or 0. Our potential solution, -16, is not among these restricted values. Therefore, it is a valid candidate for the solution. Before we definitively declare -16 as the solution, we must perform a crucial step: checking for extraneous solutions. This involves substituting our potential solution back into the original equation to ensure that it satisfies the equation and does not result in any undefined expressions. Solving the simplified equation is a critical step in the process, but it's not the final step. We must always be vigilant about the domain restrictions and the possibility of extraneous solutions. The algebraic manipulations we performed in this step were straightforward, but they rely on the fundamental principles of equation solving: maintaining balance by performing the same operation on both sides. We distributed, combined like terms, and isolated the variable, all while adhering to these principles. In the next step, we will rigorously check our potential solution to ensure its validity. This step is often overlooked, but it is essential for obtaining the correct answer. We will substitute x = -16 back into the original equation and verify that both sides are equal.

Step 4: Checking for Extraneous Solutions

The final and arguably most crucial step in solving rational equations is checking for extraneous solutions. As we've emphasized, multiplying both sides of an equation by an expression containing a variable can sometimes introduce solutions that do not satisfy the original equation. These are called extraneous solutions, and they arise because the multiplication can alter the domain of the equation. To check for extraneous solutions, we must substitute our potential solution(s) back into the original equation and verify that both sides are equal and that no denominators become zero. In our case, we found a potential solution of x = -16. Let's substitute this value into the original equation: 2/(x+8) = 4/x. Substituting x = -16, we get: 2/(-16+8) = 4/(-16). Now, let's simplify both sides: 2/(-8) = -1/4. And on the right side: 4/(-16) = -1/4. Both sides of the equation are equal: -1/4 = -1/4. This confirms that x = -16 is indeed a valid solution and does not lead to any inconsistencies. Furthermore, we must also ensure that our solution does not make any of the original denominators zero. We already know that x cannot be -8 or 0. Our solution, x = -16, is different from these restricted values, so it passes this test as well. Since x = -16 satisfies the original equation and does not violate any domain restrictions, we can confidently conclude that it is the solution to the equation. Checking for extraneous solutions is not merely a formality; it's a necessary safeguard against incorrect answers. It ensures that the solution we've found is a true solution and not an artifact of the algebraic manipulations. The process of solving rational equations involves several steps, each with its own importance. Identifying the domain, clearing the fractions, solving the simplified equation, and checking for extraneous solutions all contribute to the final result. In the next section, we will summarize the steps we've taken and present the final solution in a clear and concise manner. This will reinforce the process and provide a comprehensive overview of how to solve rational equations.

Solution

Summary and Final Answer

Having meticulously worked through each step, we arrive at the solution to the equation 2/(x+8) = 4/x. Let's recap the key steps we undertook:

  1. Identifying the Domain: We recognized the importance of domain restrictions and determined that x cannot be -8 or 0.
  2. Clearing the Fractions: We multiplied both sides of the equation by the least common denominator, x(x+8), to eliminate the fractions.
  3. Solving the Simplified Equation: We solved the resulting linear equation, 2x = 4(x+8), and found a potential solution of x = -16.
  4. Checking for Extraneous Solutions: We substituted x = -16 back into the original equation and verified that it is a valid solution.

Since x = -16 satisfies the original equation and is not a restricted value, we can confidently state the final solution. Therefore, the solution to the equation 2/(x+8) = 4/x is x = -16. This example illustrates the general strategy for solving rational equations. While the specific details may vary depending on the equation, the underlying principles remain the same. Always remember to identify the domain, clear the fractions, solve the resulting equation, and most importantly, check for extraneous solutions. By following these steps diligently, you can confidently solve a wide range of rational equations. Solving rational equations is a valuable skill in mathematics, with applications in various fields. Mastering this skill requires understanding the underlying concepts and practicing the techniques. This article has provided a detailed guide to solving one particular rational equation, but the principles discussed can be applied to many others. Remember to approach each equation systematically, paying attention to the details, and you will be well-equipped to tackle these types of problems. In conclusion, the solution to the equation 2/(x+8) = 4/x is x = -16. This solution was obtained by carefully following the steps outlined in this article, emphasizing the importance of each step in the process.