Solving Rational Equations: Find X In 3/(x+1) - 1/2 = 4/(3x+3)

by Alex Johnson 63 views

Let's dive into the world of rational equations! If you've ever felt a little intimidated by fractions and variables mixed together, don't worry. We're going to break down the process of solving the equation 3/(x+1) - 1/2 = 4/(3x+3) step by step, making it clear and easy to understand. This guide will not only help you solve this specific problem but also equip you with the skills to tackle similar equations with confidence. So, grab a pencil and paper, and let's get started!

Understanding Rational Equations

Before we jump into solving, let's make sure we're on the same page about what a rational equation actually is. Rational equations are simply equations that contain one or more rational expressions. A rational expression is a fraction where the numerator and denominator are polynomials. Think of it as an equation involving fractions with variables in the denominator. These equations often appear more complex than linear or quadratic equations, but with the right approach, they become manageable. The key to solving them lies in eliminating the fractions, which we'll do by finding a common denominator. This initial step transforms the equation into a more familiar form, making it easier to solve for the variable. Keep in mind that when dealing with rational equations, it's crucial to check for extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. This often happens when a solution makes the denominator of the original equation equal to zero, rendering the expression undefined. Therefore, after finding potential solutions, we must always plug them back into the original equation to ensure they are valid.

Step-by-Step Solution

Now, let's get to the heart of the matter: solving the equation 3/(x+1) - 1/2 = 4/(3x+3). We'll break it down into manageable steps:

1. Find the Least Common Denominator (LCD)

The first and most crucial step in solving this rational equation is to identify the least common denominator (LCD). The LCD is the smallest multiple that all the denominators in the equation divide into evenly. This will allow us to eliminate the fractions and simplify the equation. In our equation, the denominators are (x+1), 2, and (3x+3). Notice that (3x+3) can be factored as 3(x+1). This observation is key because it reveals the common factor of (x+1) between the first and third denominators. When determining the LCD, we need to consider each unique factor and its highest power present in any of the denominators. Here, the unique factors are 2, 3, and (x+1). Therefore, the LCD is the product of these factors: 2 * 3 * (x+1) = 6(x+1). Understanding how to find the LCD is fundamental to solving rational equations, as it sets the stage for simplifying the equation and isolating the variable.

2. Multiply Both Sides by the LCD

Now that we've found the LCD, the next step is to multiply both sides of the equation by it. This is a critical step because it eliminates the fractions, transforming our rational equation into a simpler, more manageable form. By multiplying each term by the LCD, we effectively clear the denominators, which makes the equation much easier to solve. In our case, the LCD is 6(x+1), so we'll multiply both sides of the equation 3/(x+1) - 1/2 = 4/(3x+3) by 6(x+1). This means we'll have to distribute 6(x+1) to each term on both sides of the equation. This process might seem a bit lengthy, but it's a direct application of the distributive property of multiplication, which states that a(b + c) = ab + ac. By carefully multiplying each term by the LCD, we ensure that the denominators will cancel out, leaving us with an equation that is free of fractions and easier to work with.

3. Simplify and Solve for x

After multiplying both sides of the equation by the LCD, our next step is to simplify the resulting expression and then solve for the variable x. This involves a combination of algebraic manipulations, such as distributing, combining like terms, and isolating x on one side of the equation. Once we've eliminated the fractions, we're typically left with a linear or quadratic equation, which we can solve using standard algebraic techniques. These techniques might include adding or subtracting terms from both sides, multiplying or dividing by constants, and, if we encounter a quadratic equation, factoring, completing the square, or using the quadratic formula. The goal is to isolate x so that we can determine its value. This step-by-step process of simplification and solving is at the heart of algebra, and it's a skill that's applicable to a wide range of mathematical problems. By carefully applying these techniques, we can systematically find the value(s) of x that satisfy the equation.

4. Check for Extraneous Solutions

Once we've found a potential solution (or solutions) for x, the final and crucial step is to check for extraneous solutions. Extraneous solutions are values that satisfy the transformed equation (after we've cleared the fractions) but do not satisfy the original rational equation. This phenomenon occurs because multiplying both sides of an equation by an expression containing a variable can introduce solutions that weren't there originally. The primary reason for extraneous solutions is that they often make the denominator of the original equation equal to zero, which is undefined in mathematics. To check for extraneous solutions, we substitute each potential solution back into the original equation and see if it holds true. If a value makes any denominator in the original equation equal to zero, it is an extraneous solution and must be discarded. This step is not just a formality; it's a critical part of the problem-solving process that ensures we have the correct and valid solutions to the equation. By diligently checking for extraneous solutions, we maintain the integrity of our solution and avoid incorrect answers.

Solving the Equation 3/(x+1) - 1/2 = 4/(3x+3) - A Detailed Walkthrough

Let's put the steps we discussed into action and solve the equation 3/(x+1) - 1/2 = 4/(3x+3). This detailed walkthrough will not only provide the solution but also reinforce the concepts and techniques we've covered. By following along, you'll gain a deeper understanding of how to tackle rational equations and the importance of each step in the process.

  1. Find the Least Common Denominator (LCD):

    • As we discussed earlier, the denominators are (x+1), 2, and (3x+3). We can factor (3x+3) as 3(x+1). Therefore, the LCD is 6(x+1).

  2. Multiply Both Sides by the LCD:

    • Multiply both sides of the equation by 6(x+1):
      6(x+1) * [3/(x+1) - 1/2] = 6(x+1) * [4/(3x+3)]
    • Distribute 6(x+1) to each term:
      6(x+1) * [3/(x+1)] - 6(x+1) * [1/2] = 6(x+1) * [4/(3(x+1))]

  3. Simplify:

    • Cancel out common factors:
      6 * 3 - 3(x+1) = 2 * 4
    • Further simplification:
      18 - 3x - 3 = 8
    • Combine like terms:
      15 - 3x = 8

  4. Solve for x:

    • Subtract 15 from both sides:
      -3x = -7
    • Divide by -3:
      x = 7/3

  5. Check for Extraneous Solutions:

    • Substitute x = 7/3 back into the original equation:
      3/((7/3)+1) - 1/2 = 4/(3(7/3)+3)
    • Simplify:
      3/(10/3) - 1/2 = 4/(7+3)

      9/10 - 1/2 = 4/10

      9/10 - 5/10 = 4/10

      4/10 = 4/10
    • The solution x = 7/3 is valid because it does not make any denominator in the original equation equal to zero.

Therefore, the solution to the equation 3/(x+1) - 1/2 = 4/(3x+3) is x = 7/3.

Common Pitfalls and How to Avoid Them

When solving rational equations, there are several common pitfalls that students often encounter. Being aware of these potential traps can help you avoid mistakes and ensure you arrive at the correct solution. One of the most frequent errors is forgetting to check for extraneous solutions. As we've emphasized, this step is crucial because multiplying both sides of an equation by an expression containing a variable can introduce values that don't actually satisfy the original equation. Another common mistake is incorrectly finding the least common denominator (LCD). A wrong LCD can lead to significant errors in the simplification process. To avoid this, always double-check that your LCD includes all unique factors from the denominators and that each factor is raised to the highest power it appears in any denominator. Additionally, errors in algebraic manipulation, such as incorrect distribution or combining like terms improperly, can also lead to wrong answers. To minimize these errors, it's a good practice to write out each step clearly and double-check your work as you go along. Lastly, be cautious when dealing with negative signs, as they are a common source of mistakes. Remember to distribute negative signs correctly and pay close attention to the signs of terms when simplifying. By being mindful of these common pitfalls and taking the time to avoid them, you can significantly improve your accuracy and confidence in solving rational equations.

Practice Problems

To truly master the art of solving rational equations, practice is essential. Working through a variety of problems will not only solidify your understanding of the steps involved but also help you develop your problem-solving skills and intuition. Start with simpler equations that have straightforward denominators and gradually progress to more complex problems with higher-degree polynomials or multiple terms. As you practice, pay attention to the common pitfalls we discussed earlier, such as forgetting to check for extraneous solutions or making mistakes in algebraic manipulation. Try to solve each problem independently, and if you get stuck, review the steps and examples we've covered. It can also be helpful to work with a study group or seek assistance from a tutor or teacher when you encounter challenging problems. Remember, the key to success in mathematics is consistent practice and a willingness to learn from your mistakes. The more you practice, the more comfortable and confident you'll become in solving rational equations. So, grab some practice problems and start honing your skills today!

Conclusion

Solving rational equations might seem daunting at first, but with a systematic approach and careful attention to detail, it becomes a manageable task. We've walked through the key steps: finding the LCD, multiplying to eliminate fractions, simplifying and solving for x, and, crucially, checking for extraneous solutions. Remember, practice makes perfect, so work through various examples to build your confidence. With consistent effort, you'll be able to tackle these equations with ease. For further learning and practice, consider exploring resources like Khan Academy's Algebra I section, which offers numerous lessons and exercises on rational expressions and equations.