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:
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For \(\frac{5}{x - 5}\): \[ \frac{5}{x - 5} = \frac{5(x + 2)}{(x - 5)(x + 2)} = \frac{5x + 10}{(x - 5)(x + 2)} \]
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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: \[ \frac{9x - 10}{(x - 5)(x + 2)} \]
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