Question

Add the rational expressions to find the sum: 2/x + 3/x+5



A
5/2x+5
B
5x+10/x^2+5x
D
13/x+5

Answer 1

To add the rational expressions \( \frac{2}{x} + \frac{3}{x+5} \), we need to find a common denominator. The denominators are \( x \) and \( x + 5 \). The common denominator will be \( x(x + 5) \).

Next, we rewrite each fraction with the common denominator:

1. For \( \frac{2}{x} \):
\[
\frac{2}{x} = \frac{2(x + 5)}{x(x + 5)} = \frac{2x + 10}{x(x + 5)}
\]

2. For \( \frac{3}{x + 5} \):
\[
\frac{3}{x + 5} = \frac{3x}{x(x + 5)} = \frac{3x}{x(x + 5)}
\]

Now, we can add them together:
\[
\frac{2x + 10}{x(x + 5)} + \frac{3x}{x(x + 5)} = \frac{2x + 10 + 3x}{x(x + 5)} = \frac{5x + 10}{x(x + 5)}
\]

So, the sum is:
\[
\frac{5x + 10}{x(x + 5)}
\]

Now, let's simplify it:
\[
\frac{5(x + 2)}{x(x + 5)}
\]

Now, let's compare this result with the provided options:

A) \( \frac{5}{2x+5} \)

B) \( \frac{5x+10}{x^2+5x} \)

C) (Not provided)

D) \( \frac{13}{x+5} \)

The correct option is **B**: \( \frac{5x + 10}{x^2 + 5x} \) since the denominator \( x^2 + 5x \) can be factored into \( x(x + 5) \).

Thus, the final answer is:
**B** \( \frac{5x + 10}{x^2 + 5x} \)