Question

Add the rational expressions and simplify.

fraction with numerator 5 and denominator x minus 5 plus fraction with numerator 4 and denominator x plus 2

Answer 1

To add the rational expressions \(\frac{5}{x - 5}\) and \(\frac{4}{x + 2}\), we need to find a common denominator. The common denominator will be the product of the individual denominators, which is \((x - 5)(x + 2)\).

Now we can rewrite each fraction with the common denominator:

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

2. For \(\frac{4}{x + 2}\):
\[
\frac{4}{x + 2} = \frac{4(x - 5)}{(x + 2)(x - 5)} = \frac{4x - 20}{(x - 5)(x + 2)}
\]

Now, we can add these two fractions together:
\[
\frac{5x + 10}{(x - 5)(x + 2)} + \frac{4x - 20}{(x - 5)(x + 2)} = \frac{(5x + 10) + (4x - 20)}{(x - 5)(x + 2)}
\]

Combine the numerators:
\[
5x + 10 + 4x - 20 = 9x - 10
\]

So, we have:
\[
\frac{9x - 10}{(x - 5)(x + 2)}
\]

Thus, the sum of the rational expressions \(\frac{5}{x - 5} + \frac{4}{x + 2}\) simplified is:
\[
\frac{9x - 10}{(x - 5)(x + 2)}
\]