Question
Add the rational expressions to find the sum: 2/x+3/x+5
5x+10/x^2+5x
13/x+5
5/2x+5
2x+8/x^2+5x
5x+10/x^2+5x
13/x+5
5/2x+5
2x+8/x^2+5x
Answers
GPT-4o mini
To find the sum of the rational expressions \( \frac{2}{x} + \frac{3}{x} + 5 \), we first combine \( \frac{2}{x} \) and \( \frac{3}{x} \).
1. **Combine the fractions with the same denominator**:
\[
\frac{2}{x} + \frac{3}{x} = \frac{2 + 3}{x} = \frac{5}{x}
\]
2. **Now add \( 5 \) to this sum**:
To add \( \frac{5}{x} \) and \( 5 \), we need to express \( 5 \) as a fraction with \( x \) as the denominator:
\[
5 = \frac{5x}{x}
\]
Now we can add \( \frac{5}{x} \) and \( \frac{5x}{x} \):
\[
\frac{5}{x} + \frac{5x}{x} = \frac{5 + 5x}{x} = \frac{5x + 5}{x}
\]
Thus, the sum of the original rational expressions is:
\[
\frac{5x + 5}{x}
\]
Now, after our calculations, let's check if that's among the provided answer choices:
- \( \frac{5x + 10}{x^2 + 5x} \) can be simplified to \( \frac{5(x + 2)}{x(x + 5)} \) if possible, but it does not match.
- \( \frac{13}{x + 5} \) is not relevant in the context.
- \( \frac{5}{2x + 5} \) does not connect to our work either.
- \( \frac{2x + 8}{x^2 + 5x} \) again doesn't match.
None of the provided options seem to match our final expression of \( \frac{5(x + 1)}{x} \).
Thus, the result of \( \frac{2}{x} + \frac{3}{x} + 5 \) simplifies to \( \frac{5(x + 1)}{x} \) or \( 5 + \frac{5}{x} \), and none of the answers are correct.
1. **Combine the fractions with the same denominator**:
\[
\frac{2}{x} + \frac{3}{x} = \frac{2 + 3}{x} = \frac{5}{x}
\]
2. **Now add \( 5 \) to this sum**:
To add \( \frac{5}{x} \) and \( 5 \), we need to express \( 5 \) as a fraction with \( x \) as the denominator:
\[
5 = \frac{5x}{x}
\]
Now we can add \( \frac{5}{x} \) and \( \frac{5x}{x} \):
\[
\frac{5}{x} + \frac{5x}{x} = \frac{5 + 5x}{x} = \frac{5x + 5}{x}
\]
Thus, the sum of the original rational expressions is:
\[
\frac{5x + 5}{x}
\]
Now, after our calculations, let's check if that's among the provided answer choices:
- \( \frac{5x + 10}{x^2 + 5x} \) can be simplified to \( \frac{5(x + 2)}{x(x + 5)} \) if possible, but it does not match.
- \( \frac{13}{x + 5} \) is not relevant in the context.
- \( \frac{5}{2x + 5} \) does not connect to our work either.
- \( \frac{2x + 8}{x^2 + 5x} \) again doesn't match.
None of the provided options seem to match our final expression of \( \frac{5(x + 1)}{x} \).
Thus, the result of \( \frac{2}{x} + \frac{3}{x} + 5 \) simplifies to \( \frac{5(x + 1)}{x} \) or \( 5 + \frac{5}{x} \), and none of the answers are correct.