Question

Add the rational expressions to find the sum: 5/x+2+6/x−3
(1 point)
Responses

11x+27x2−x−6
the fraction with numerator 11 x plus 27 and denominator x squared minus x minus 6

11x−3x2−x−6
the fraction with numerator 11 x minus 3 and denominator x squared minus x minus 6

−3x−6
the fraction with numerator negative 3 and denominator x minus 6

11x−1

Answer 1

To find the sum of the rational expressions \( \frac{5}{x+2} + \frac{6}{x-3} \), we first need to find a common denominator. The denominators in this case are \( x + 2 \) and \( x - 3 \). The common denominator will be \( (x + 2)(x - 3) \).

Now, we can rewrite each fraction with this common denominator:

1. For the first fraction \( \frac{5}{x+2} \):
\[
\frac{5}{x+2} \cdot \frac{x-3}{x-3} = \frac{5(x-3)}{(x+2)(x-3)} = \frac{5x - 15}{(x+2)(x-3)}
\]

2. For the second fraction \( \frac{6}{x-3} \):
\[
\frac{6}{x-3} \cdot \frac{x+2}{x+2} = \frac{6(x+2)}{(x-3)(x+2)} = \frac{6x + 12}{(x-3)(x+2)}
\]

Now we can add these two fractions:
\[
\frac{5x - 15 + 6x + 12}{(x+2)(x-3)} = \frac{(5x + 6x) + (-15 + 12)}{(x+2)(x-3)} = \frac{11x - 3}{(x+2)(x-3)}
\]

Next, we need to simplify the denominator. The product of \( x + 2 \) and \( x - 3 \) gives us:
\[
(x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6
\]

Thus, the overall expression simplifies to:
\[
\frac{11x - 3}{x^2 - x - 6}
\]

Now we can match this expression with the provided choices. The relevant answer is:
\[
\text{11x−3/ x²−x−6}
\]

So, the correct answer is:
**11x−3x²−x−6**.