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Solve the following mathematical problems:

simplify: $\frac{x - 1}{x^{2} - x - 20} + \frac{4 - x}{x^{2} - 4 x - 5}$ (Equation)

dsuc.created: 2 years ago | dsuc.updated: 2 years ago

dsuc.updated: 2 years ago

$\frac{x - 1}{x^{2} - x - 20} + \frac{4 - x}{x^{2} - 4 x - 5} = \frac{x - 1}{x^{2} - 5 x + 4 x - 20} + \frac{4 - x}{x^{2} - 5 x + x - 5} = \frac{x - 1}{x (x - 5) + 4 (x - 5)} + \frac{4 - x}{x (x - 5) + 1 (x - 5)} = \frac{x - 1}{(x - 5) (x + 4)} + \frac{4 - x}{(x - 5) (x + 1)} = \frac{(x - 5) (x + 1) + (4 - x) (x + 4)}{(x - 5) (x + 4) (x + 1)} = \frac{(x - 1) (x + 1) + (4 + x) (4 - x)}{(x - 5) (x + 4) (x + 1)} = \frac{x^{2} - 1 + (4^{2} - x^{2})}{(x - 5) (x + 4) (x + 1)} = \frac{x^{2} - 1 + 16 - x^{2}}{(x - 5) (x + 4) (x + 1)} = \frac{15}{(x - 5) (x + 4) (x + 1)}$