Simplify The Expression 3/x + 2/(x+4).

In the realm of algebra, a fundamental skill lies in the ability to manipulate and simplify expressions, especially those involving fractions. This article delves into the process of adding two algebraic fractions, specifically $\frac{3}{x}$\frac{3}{x}} and ${\frac{2$x+4}}. Mastering this technique is crucial for tackling more complex algebraic problems and is a cornerstone of mathematical proficiency. We will break down the steps involved, ensuring a clear and comprehensive understanding for anyone looking to enhance their algebraic skills. Understanding how to add fractions with variables is a fundamental concept in algebra. This guide provides a clear, step-by-step approach to simplifying the expression ${\frac{3{x} + \frac{2}{x+4}}$, ensuring you grasp the core principles involved. Algebraic fractions, much like their numerical counterparts, require a common denominator before they can be added. This process involves finding the least common multiple (LCM) of the denominators and adjusting the numerators accordingly. The final step is to simplify the resulting fraction, which may involve combining like terms and factoring. By the end of this guide, you will have a solid understanding of how to add these types of fractions and be well-equipped to tackle similar problems. Remember, practice is key to mastering these skills, so don't hesitate to work through additional examples and challenge yourself with more complex expressions. This foundation will prove invaluable as you progress in your mathematical journey.

1. Finding the Least Common Denominator (LCD)

To successfully add fractions, a common denominator is paramount. The least common denominator (LCD) is the smallest multiple that both denominators share. In our case, the denominators are x and (x+4). Since these expressions do not share any common factors, their LCD is simply their product:

LCD = x(x+4)

Finding the least common denominator (LCD) is the first crucial step in adding algebraic fractions. The LCD is the smallest expression that is divisible by both denominators. In our case, the denominators are x and (x + 4). Since these expressions are distinct and share no common factors, the LCD is simply their product. Understanding how to find the LCD is essential, as it allows us to rewrite the fractions with a common base, making addition possible. This step is analogous to finding a common denominator when adding numerical fractions, but it involves algebraic expressions instead of numbers. Once you've identified the LCD, you can proceed to rewrite each fraction with this new denominator. This involves multiplying both the numerator and the denominator of each fraction by the factor that makes its denominator equal to the LCD. This process ensures that the value of the fraction remains unchanged while facilitating the addition. Mastering the concept of the LCD is a cornerstone of algebraic fraction manipulation, paving the way for more complex operations and problem-solving in algebra. By taking the time to understand this step thoroughly, you'll build a solid foundation for your future mathematical endeavors.

2. Rewriting the Fractions

Now, we rewrite each fraction using the LCD. For the first fraction, ${\frac{3}{x}}$, we multiply both the numerator and denominator by (x+4):

${\frac{3}{x} \times \frac{x+4}{x+4} = \frac{3(x+4)}{x(x+4)}}$

For the second fraction, ${\frac{2}{x+4}}$, we multiply both the numerator and denominator by x:

${\frac{2}{x+4} \times \frac{x}{x} = \frac{2x}{x(x+4)}}$

Rewriting the fractions with the least common denominator (LCD) is a critical step in the addition process. This step ensures that we can combine the fractions because they now have the same denominator. For the first fraction, ${\frac{3}{x}}$, we identified that we need to multiply both the numerator and the denominator by (x + 4) to achieve the LCD of x(x + 4). This multiplication yields ${\frac{3(x+4)}{x(x+4)}}$. Similarly, for the second fraction, ${\frac{2}{x+4}}$, we multiply both the numerator and the denominator by x to obtain the LCD. This gives us ${\frac{2x}{x(x+4)}}$. Rewriting fractions in this manner maintains their original values while allowing us to perform the addition. The key is to multiply both the numerator and the denominator by the same expression, which is equivalent to multiplying by 1, thus preserving the fraction's value. This technique is fundamental in working with algebraic fractions and is a skill that will be applied repeatedly in more advanced algebraic manipulations. By mastering this step, you'll be well-prepared to add, subtract, and simplify complex expressions involving fractions. Remember to always double-check your work to ensure that you've correctly multiplied both the numerator and the denominator, as any errors at this stage will propagate through the rest of the solution.

3. Adding the Fractions

With a common denominator, we can now add the fractions by adding the numerators:

${\frac{3(x+4)}{x(x+4)} + \frac{2x}{x(x+4)} = \frac{3(x+4) + 2x}{x(x+4)}}$

Now, we simplify the numerator:

${\frac{3x + 12 + 2x}{x(x+4)} = \frac{5x + 12}{x(x+4)}}$

Adding the fractions is the core step once we have a common denominator. This involves combining the numerators while keeping the denominator the same. In our case, we have ${\frac{3(x+4)}{x(x+4)} + \frac{2x}{x(x+4)}}$. We add the numerators together to get ${\frac{3(x+4) + 2x}{x(x+4)}}$. The next crucial step is to simplify the resulting expression. Simplifying the numerator often involves distributing and combining like terms. In our example, we distribute the 3 in the expression 3(x + 4), which gives us 3x + 12. Adding this to 2x results in 3x + 12 + 2x. Combining the like terms (3x and 2x), we get 5x + 12. Therefore, the expression becomes ${\frac{5x + 12}{x(x+4)}}$. This process of simplifying the numerator is essential for obtaining the final, most concise form of the fraction. It's important to be meticulous in this step, ensuring that all terms are correctly combined and that no arithmetic errors are made. By carefully simplifying the numerator, we pave the way for the final simplification steps, which may involve factoring or other algebraic manipulations. Mastering this process is a key component of working with algebraic fractions and will be invaluable in tackling more complex problems.

4. Simplifying the Result

Finally, we check if the resulting fraction, ${\frac{5x + 12}{x(x+4)}}$, can be simplified further. In this case, the numerator (5x + 12) does not share any common factors with the denominator x(x+4). Therefore, the fraction is already in its simplest form.

The final step in simplifying algebraic fractions is to ensure that the resulting fraction is in its most reduced form. This involves checking for any common factors between the numerator and the denominator that can be canceled out. In our case, we have the fraction ${\frac{5x + 12}{x(x+4)}}$. Simplifying fractions requires a careful examination of both the numerator and the denominator. We need to determine if there are any factors that are present in both expressions. In this specific example, the numerator is 5x + 12, which is a linear expression. The denominator is x(x + 4), which is a quadratic expression when expanded. We look for any common factors between 5x + 12 and x or (x + 4). Since 5x + 12 cannot be factored further and it doesn't share any factors with x or (x + 4), we conclude that the fraction is already in its simplest form. If we had found a common factor, we would have divided both the numerator and the denominator by that factor to reduce the fraction. This process of checking for common factors and canceling them out is crucial for expressing the fraction in its most simplified form. It ensures that the final answer is as concise and clear as possible. Mastering this step is essential for achieving accuracy and efficiency in algebraic manipulations.

Final Answer

Therefore, the simplified form of ${\frac{3}{x} + \frac{2}{x+4}}$ is:

${\frac{5x + 12}{x(x+4)}}$

In conclusion, we have successfully added and simplified the algebraic fractions ${\frac{3}{x}}$ and ${\frac{2}{x+4}}$. The process involved finding the least common denominator (LCD), rewriting the fractions with the LCD, adding the numerators, simplifying the resulting expression, and finally, ensuring the fraction was in its simplest form. The final answer, ${\frac{5x + 12}{x(x+4)}}$, represents the sum of the two fractions in its most reduced form. This exercise highlights the importance of mastering the fundamental principles of algebraic fraction manipulation. These skills are essential for success in higher-level mathematics and are frequently applied in various fields, including engineering, physics, and computer science. By understanding the steps involved in adding and simplifying algebraic fractions, you gain a valuable tool for solving complex problems and a deeper appreciation for the elegance of algebraic expressions. Remember, practice is key to mastery, so continue to work through similar problems to solidify your understanding and build your confidence. The ability to manipulate algebraic fractions is a cornerstone of mathematical proficiency, and this guide provides a solid foundation for your future endeavors.