Simplify 3/(2x+5) + 5/(x-5) A Step-by-Step Guide

In the realm of algebra, simplifying expressions is a fundamental skill. One common task involves adding rational expressions, which are fractions containing polynomials. This article delves into the process of simplifying the expression $\frac{3}{2x+5} + \frac{5}{x-5}$, providing a step-by-step guide and offering insights into the underlying concepts. Understanding how to simplify rational expressions is crucial for various mathematical applications, from solving equations to analyzing functions. This guide will not only walk you through the solution but also ensure you grasp the core principles, making you proficient in handling similar problems.

Understanding Rational Expressions

Rational expressions, at their core, are fractions where the numerator and denominator are polynomials. Just like regular fractions, we can perform operations such as addition, subtraction, multiplication, and division on them. However, the presence of variables adds a layer of complexity. To effectively manipulate rational expressions, it's essential to understand the concept of a common denominator. When adding or subtracting fractions, a common denominator is necessary to combine the terms. This principle holds true for rational expressions as well. Finding the least common denominator (LCD) simplifies the process and helps avoid unnecessary complications. The LCD is the smallest multiple that both denominators share, ensuring the fractions can be combined efficiently. In this article, we will specifically focus on the addition of rational expressions, highlighting the steps involved in finding the LCD and combining the fractions.

The Importance of a Common Denominator

When adding or subtracting fractions, whether they are simple numerical fractions or complex rational expressions, the common denominator plays a pivotal role. Think of it as the common language that allows us to combine different fractional parts into a unified whole. Without a common denominator, we would be attempting to add unlike quantities, much like trying to add apples and oranges. The common denominator provides a standardized unit, allowing us to accurately combine the numerators. In the context of rational expressions, the denominators often involve variables, making the process of finding a common denominator slightly more intricate. However, the underlying principle remains the same: we need a denominator that is divisible by both original denominators. This ensures that we can rewrite each fraction with the same denominator, making addition or subtraction a straightforward process. The least common denominator (LCD) is the most efficient choice, as it simplifies the resulting expression and avoids unnecessary complexity.

Identifying the Least Common Denominator (LCD)

To successfully add rational expressions, the first crucial step is identifying the least common denominator (LCD). The LCD is the smallest expression that is divisible by both denominators. For numerical fractions, finding the LCD often involves listing multiples or using prime factorization. With rational expressions, the process is similar but involves factoring polynomials. Start by examining the denominators and factoring them completely. This means breaking down each polynomial into its irreducible factors. Once the denominators are factored, the LCD is formed by taking the highest power of each unique factor present in either denominator. For instance, if one denominator has a factor of $(x+2)$ and the other has $(x+2)^2$, the LCD will include $(x+2)^2$. In our specific problem, the denominators are $(2x+5)$ and $(x-5)$. Since these are linear expressions and cannot be factored further, the LCD is simply their product: $(2x+5)(x-5)$. Understanding how to identify the LCD is fundamental to simplifying rational expressions and is a skill that will be used repeatedly in algebra and calculus.

Step-by-Step Solution

Now, let's walk through the process of simplifying the given expression: $\frac{3}{2x+5} + \frac{5}{x-5}$. This step-by-step solution will not only provide the answer but also reinforce the concepts discussed earlier.

Step 1: Find the Least Common Denominator (LCD)

As discussed earlier, the denominators are $(2x+5)$ and $(x-5)$. These are both linear expressions and cannot be factored further. Therefore, the least common denominator (LCD) is simply their product:

LCD = $(2x+5)(x-5)$

This LCD will serve as the common denominator for both fractions, allowing us to combine them.

Step 2: Rewrite the Fractions with the LCD

Next, we need to rewrite each fraction with the LCD as the denominator. This involves multiplying the numerator and denominator of each fraction by the appropriate factor. For the first fraction, $\frac{3}{2x+5}$, we need to multiply both the numerator and denominator by $(x-5)$:

$\frac{3}{2x+5} \cdot \frac{x-5}{x-5} = \frac{3(x-5)}{(2x+5)(x-5)}$

For the second fraction, $\frac{5}{x-5}$, we need to multiply both the numerator and denominator by $(2x+5)$:

$\frac{5}{x-5} \cdot \frac{2x+5}{2x+5} = \frac{5(2x+5)}{(2x+5)(x-5)}$

Now, both fractions have the same denominator, allowing us to proceed with addition.

Step 3: Add the Numerators

With a common denominator in place, we can now add the numerators. This involves combining the expressions in the numerators while keeping the denominator the same:

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

Next, we need to simplify the numerator by distributing and combining like terms.

Step 4: Simplify the Numerator

To simplify the numerator, we first distribute the constants:

$3(x-5) + 5(2x+5) = 3x - 15 + 10x + 25$

Now, we combine like terms:

$3x - 15 + 10x + 25 = 13x + 10$

So, the simplified numerator is $13x + 10$.

Step 5: Write the Simplified Expression

Finally, we write the simplified expression by placing the simplified numerator over the common denominator:

$\frac{13x + 10}{(2x+5)(x-5)}$

This is the simplified form of the original expression.

Analyzing the Result

The simplified expression, $\frac{13x + 10}{(2x+5)(x-5)}$, reveals the result of adding the two rational expressions. The numerator, $13x + 10$, is a linear expression, while the denominator, $(2x+5)(x-5)$, is a quadratic expression when expanded. It's important to note that the denominator cannot be zero, as this would make the expression undefined. Therefore, $x$ cannot be equal to $-\frac{5}{2}$ or $5$. This understanding of the domain is crucial when working with rational expressions. By simplifying the expression, we have made it easier to analyze and use in further calculations. For instance, if we needed to solve an equation involving this expression, the simplified form would be much easier to work with. Additionally, this simplified form allows us to identify any potential discontinuities or asymptotes in the graph of the corresponding rational function.

Checking for Further Simplification

After simplifying a rational expression, it's always a good practice to check if further simplification is possible. This often involves looking for common factors between the numerator and denominator that can be canceled out. In our case, the numerator is $13x + 10$, and the denominator is $(2x+5)(x-5)$. There are no common factors between these expressions, meaning that the expression is indeed in its simplest form. However, in other scenarios, you might encounter situations where the numerator and denominator share a common factor. In such cases, factoring both the numerator and denominator and canceling out the common factors is necessary to achieve the simplest form. This step ensures that the expression is represented in its most concise and manageable form.

Common Mistakes to Avoid

When simplifying rational expressions, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and ensure accuracy. One frequent error is failing to find a common denominator before adding or subtracting fractions. This is a fundamental step, and skipping it will inevitably lead to an incorrect result. Another common mistake is incorrectly distributing constants or signs when simplifying the numerator. Pay close attention to the distributive property and ensure that you are applying it correctly. Additionally, be cautious when canceling factors. You can only cancel factors that are multiplied, not terms that are added or subtracted. For example, you cannot cancel the $x$ in $\frac{x+2}{x}$ because the $x$ in the numerator is part of a sum. Finally, always remember to check for further simplification after you have combined the fractions. This ensures that your answer is in its simplest form.

The Importance of Careful Distribution

A critical step in simplifying rational expressions, especially when adding or subtracting them, is the distribution of constants or signs. Errors in distribution are among the most common mistakes and can lead to incorrect results. When you multiply a constant by a polynomial, ensure that you multiply the constant by every term within the polynomial. For example, in the expression $3(x-5)$, you must multiply both $x$ and $-5$ by $3$, resulting in $3x - 15$. Similarly, when subtracting a polynomial, remember to distribute the negative sign to every term. For instance, if you have $-(2x+5)$, it becomes $-2x - 5$. Careless distribution can easily lead to sign errors or incorrect coefficients, which will propagate through the rest of the solution. To avoid these mistakes, double-check your work and be meticulous in applying the distributive property. This attention to detail will significantly improve your accuracy when simplifying rational expressions.

Conclusion

In conclusion, simplifying rational expressions involves a series of steps, each crucial to arriving at the correct answer. The process begins with identifying the least common denominator (LCD), which allows us to rewrite the fractions with a common base. We then add the numerators, simplify the resulting expression, and check for further simplification. Understanding the underlying principles, such as the importance of a common denominator and careful distribution, is key to mastering this skill. By avoiding common mistakes and practicing regularly, you can become proficient in simplifying rational expressions. This skill is not only essential for algebra but also forms a foundation for more advanced mathematical concepts. The ability to manipulate and simplify rational expressions opens doors to solving complex equations, analyzing functions, and tackling real-world problems involving ratios and proportions. This article has provided a comprehensive guide to simplifying the expression $\frac{3}{2x+5} + \frac{5}{x-5}$, equipping you with the knowledge and skills to confidently approach similar problems in the future. Remember, practice makes perfect, so continue to challenge yourself with various examples to solidify your understanding.

Answer: B) $\frac{13x+10}{(x-5)(2x+5)}$