[(x/x+3)+(3/x+3)+(2/x+3)] sum

[(x/x+3)+(3/x+3)+(2/x+3)] sum

The Correct Answer and Explanation is :

To find the sum of the expression (\left(\frac{x}{x+3}\right) + \left(\frac{3}{x+3}\right) + \left(\frac{2}{x+3}\right)), we will first combine the terms, taking care to find a common denominator.

Step 1: Identify the Common Denominator

The common denominator for all three fractions is (x + 3).

Step 2: Rewrite Each Term

The first term can be rewritten as:

[
\frac{x}{x+3} = \frac{x}{x+3}
]

The second term becomes:

[
\frac{3}{x+3} = \frac{3}{x+3}
]

The third term is:

[
\frac{2}{x+3} = \frac{2}{x+3}
]

Step 3: Combine the Fractions

Now that we have a common denominator, we can combine the fractions:

[
\frac{x}{x+3} + \frac{3}{x+3} + \frac{2}{x+3} = \frac{x + 3 + 2}{x + 3}
]

Step 4: Simplify the Numerator

Now, combine the terms in the numerator:

[
x + 3 + 2 = x + 5
]

Thus, the combined expression becomes:

[
\frac{x + 5}{x + 3}
]

Step 5: Final Answer

So, the sum of the expression is:

[
\frac{x + 5}{x + 3}
]

Explanation

In this calculation, we employed basic algebraic principles to manipulate fractions. The main concept is finding a common denominator, which allows us to combine multiple fractions into one. The individual terms were structured similarly, sharing the denominator (x + 3), which simplified the addition process.

After rewriting the fractions over the common denominator, we combined the numerators, simplifying it to a single fraction that maintains the relationship established by the original terms. This method of fraction addition is vital in algebra, particularly when dealing with rational expressions, as it lays the groundwork for more complex operations such as solving equations or integrating functions.

Thus, understanding these foundational skills is crucial for progressing in algebra and calculus.