What is the sum? x/x+3 + 3/x+3 + 2/x+3
A) 5/3
B) x+5/x+3
C) x+5/3x+27
D) 6x/x+3
The Correct Answer and Explanation is :
The given expression is:
[
\frac{x}{x+3} + \frac{3}{x+3} + \frac{2}{x+3}
]
Step 1: Combine the fractions
Since all three fractions have the same denominator, we can combine them by adding their numerators:
[
\frac{x}{x+3} + \frac{3}{x+3} + \frac{2}{x+3} = \frac{x + 3 + 2}{x+3}
]
Step 2: Simplify the numerator
Simplifying the numerator:
[
x + 3 + 2 = x + 5
]
Thus, the expression becomes:
[
\frac{x + 5}{x + 3}
]
Step 3: Match the result with the options
The expression we obtained is:
[
\frac{x + 5}{x + 3}
]
This corresponds to Option B:
[
\frac{x + 5}{x + 3}
]
Explanation (300 words):
In algebra, adding rational expressions follows similar rules as adding fractions. When adding fractions with the same denominator, you can simply add the numerators together while keeping the denominator unchanged. This is the approach we used in solving the given expression.
We started with three terms: (\frac{x}{x+3}), (\frac{3}{x+3}), and (\frac{2}{x+3}), all of which had the common denominator (x+3). Since the denominators were the same, we combined the numerators (x), (3), and (2) into a single expression:
[
x + 3 + 2 = x + 5
]
After combining the numerators, the resulting expression became:
[
\frac{x+5}{x+3}
]
This is a simplified form of the sum of the three rational expressions. The key idea is recognizing the common denominator and correctly summing the numerators while leaving the denominator as is. Finally, we compared our simplified result with the provided answer choices and found that Option B: (\frac{x + 5}{x + 3}) is the correct answer.
This process helps us understand how to work with rational expressions, a fundamental algebraic concept useful in more complex mathematical problems.