Adding Rational Expressions: 6/(x-4) + 5/x Solution

Have you ever wondered how to add fractions when they involve variables? Adding rational expressions might seem daunting at first, but with a clear, step-by-step approach, it becomes a manageable task. This article will guide you through the process of finding the sum of two rational expressions: 6/(x-4) and 5/x. We'll break down each step, ensuring you understand the underlying concepts and can confidently tackle similar problems. So, let's dive in and unravel the mystery of adding rational expressions!

Understanding Rational Expressions

Before we jump into the solution, let's define what rational expressions are. Rational expressions are essentially fractions where the numerator and the denominator are polynomials. Polynomials, in turn, are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Examples of polynomials include x^2 + 3x - 2 and 5x^4 - 7. Therefore, a rational expression looks like P(x)/Q(x), where P(x) and Q(x) are polynomials, and Q(x) is not equal to zero (because division by zero is undefined).

In our specific problem, we have two rational expressions: 6/(x-4) and 5/x. Here, 6 and 5 can be considered constant polynomials, and (x-4) and x are linear polynomials. Adding these expressions requires finding a common denominator, a crucial step that ensures we're combining like terms, much like when adding regular numerical fractions. The process of finding this common denominator and simplifying the result is the core of this article.

Why is understanding rational expressions so important? Well, these expressions appear in various areas of mathematics and physics, from calculus to circuit analysis. Being able to manipulate them, including addition, subtraction, multiplication, and division, is a fundamental skill. This article focuses on addition, but the principles we cover lay the groundwork for tackling other operations as well. Understanding the structure and behavior of rational expressions allows you to solve more complex equations and understand real-world phenomena modeled by these expressions.

Step 1: Finding the Least Common Denominator (LCD)

The first crucial step in adding rational expressions is to find the least common denominator (LCD). The LCD is the smallest expression that is divisible by both denominators in our given rational expressions. It's the equivalent of finding the least common multiple (LCM) when dealing with numerical fractions. In our case, the denominators are (x-4) and x. Since these two expressions don't share any common factors, the LCD is simply their product. The LCD is x * (x - 4), which we can also write as x(x - 4).

Why is the LCD so important? Think of adding fractions like 1/3 and 1/4. You can't directly add them because they have different denominators. You need a common denominator, which in this case is 12. You then rewrite each fraction with the denominator of 12 (4/12 and 3/12) before adding the numerators. The same principle applies to rational expressions. Finding the LCD allows us to rewrite the expressions with a common base, making addition possible. It ensures that we are adding comparable quantities, just as the common denominator in numerical fractions ensures we are adding 'parts of the same whole'.

To solidify your understanding, consider what would happen if we didn't use the LCD. We could, for instance, use any common multiple of the denominators, such as 2x(x-4) or even x^2(x-4). However, using a larger common denominator means more work simplifying the expression later. The LCD, being the least common denominator, minimizes the complexity of the final simplification. It's the most efficient path to the correct answer. Mastering the art of finding the LCD not only simplifies the addition process but also helps in more advanced algebraic manipulations involving rational expressions.

Step 2: Rewriting the Rational Expressions

Now that we've found the LCD, the next step is to rewrite each rational expression so that it has the LCD as its denominator. This involves multiplying both the numerator and the denominator of each expression by a suitable factor. Remember, multiplying both the top and bottom of a fraction by the same value doesn't change the fraction's overall value; it's like multiplying by 1.

For the first expression, 6/(x-4), we need to multiply the denominator (x-4) by x to get the LCD, x(x-4). So, we also multiply the numerator, 6, by x. This gives us (6 * x) / [x(x-4)], which simplifies to 6x / [x(x-4)]. For the second expression, 5/x, we need to multiply the denominator x by (x-4) to get the LCD. Similarly, we multiply the numerator, 5, by (x-4). This results in [5 * (x-4)] / [x(x-4)], which simplifies to 5(x-4) / [x(x-4)].

Why is this step crucial? Because it transforms the original expressions into equivalent fractions that share a common denominator. This common denominator allows us to directly add the numerators, which is the next step in our process. Without this rewriting, we'd be trying to add 'apples and oranges' – fractions with different denominators that can't be directly combined. This step ensures that we're adding like terms, a fundamental principle in algebraic manipulations.

Think of it like converting measurements. You can't directly add inches and feet until you convert them to the same unit, say inches. Similarly, rewriting rational expressions with a common denominator allows us to express them in 'the same unit', making addition a straightforward process. By mastering this technique, you'll be able to confidently handle more complex algebraic problems involving fractions and rational functions. The ability to rewrite expressions into equivalent forms is a cornerstone of algebraic manipulation, and this step provides valuable practice in that skill.

Step 3: Adding the Numerators

With both rational expressions now sharing the same denominator, it's time to add the numerators. This is where the fractions finally combine into a single expression. We simply add the numerators together, keeping the common denominator the same. From the previous step, we have 6x / [x(x-4)] and 5(x-4) / [x(x-4)]. Adding the numerators gives us (6x + 5(x-4)) / [x(x-4)].

Now, we need to simplify the numerator. First, distribute the 5 in the term 5(x-4), which gives us 5x - 20. So, the numerator becomes 6x + 5x - 20. Combining like terms (6x and 5x), we get 11x - 20. Therefore, the sum of the numerators is 11x - 20, and our combined rational expression is (11x - 20) / [x(x-4)].

Why is this step so important? It's where the actual addition takes place. Once the fractions have a common denominator, adding them becomes as simple as adding their numerators. This is a direct application of the principle of fraction addition, which states that a/c + b/c = (a+b)/c. By performing this step correctly, we're effectively combining the two rational expressions into a single, equivalent expression. This simplification is a key goal in many algebraic manipulations.

Imagine trying to combine two pieces of information that are expressed in different units. You'd first need to convert them to a common unit before adding them. Similarly, adding the numerators after finding a common denominator is like combining information expressed in the same 'fractional unit'. This step highlights the power of common denominators in simplifying complex expressions. It's a foundational skill for handling more advanced topics in algebra and calculus, such as partial fraction decomposition and integration of rational functions. Understanding how to correctly add numerators is essential for successfully navigating these concepts.

Step 4: Simplifying the Result

The final step in adding rational expressions is to simplify the resulting expression. This often involves expanding the denominator (if it's factored), combining like terms in the numerator, and, most importantly, looking for common factors that can be canceled out between the numerator and the denominator. This cancellation is crucial for expressing the result in its simplest form.

From the previous step, we have (11x - 20) / [x(x-4)]. Let's first expand the denominator: x(x-4) becomes x^2 - 4x. So, our expression is now (11x - 20) / (x^2 - 4x). Now, we look for common factors between the numerator and the denominator. In this case, there are no common factors that can be canceled out. The numerator, 11x - 20, and the denominator, x^2 - 4x, do not share any factors that would allow for simplification through cancellation. Therefore, the simplified form of our sum is (11x - 20) / (x^2 - 4x).

Why is simplifying so important? It ensures that your answer is presented in the most concise and understandable form. An unsimplified expression might be technically correct, but it's like giving directions that are unnecessarily complex when a simpler route exists. Simplifying makes the expression easier to work with in subsequent calculations and provides a clearer understanding of the expression's behavior. In the context of rational expressions, simplification often involves canceling common factors, which can significantly reduce the complexity of the expression.

Moreover, simplifying helps prevent errors in future steps. A complex, unsimplified expression can easily lead to mistakes in further calculations, such as solving equations or graphing functions. By simplifying at each step, you minimize the risk of carrying errors forward. In our example, although we couldn't cancel any factors, recognizing that the expression is already in its simplest form is a crucial part of the simplification process. This understanding prevents unnecessary attempts at factorization or cancellation, saving time and effort. Mastering the art of simplification is essential for achieving accuracy and efficiency in algebraic manipulations and is a skill that pays dividends in more advanced mathematical studies.

Conclusion

In this article, we've walked through the process of adding rational expressions, using the example of 6/(x-4) + 5/x. We covered the crucial steps of finding the least common denominator, rewriting the expressions, adding the numerators, and simplifying the result. By mastering these steps, you can confidently tackle similar problems and build a strong foundation for more advanced algebraic concepts. Remember, practice is key to solidifying your understanding, so try working through additional examples.

If you're interested in learning more about rational expressions and other algebraic concepts, a great resource is Khan Academy's Algebra I section, which offers free lessons and practice exercises. Keep exploring and expanding your mathematical skills!