Question

1/x-3 - 1/x-2 = 1/x-6 - 1/x-5

Answer 1

Resposta:

x = 4

$\frac{1}{x - 3} - \frac{1}{x - 2} = \frac{1}{x - 6} - \frac{1}{x - 5} \\ \\ faz \: o \: mmc \: de \: x - 3 \: e \: x - 2 \\ \\ \frac{(x - 2 )}{(x - 3)(x - 2)} - \frac{(x - 3)}{(x - 2)(x - 3)} = \frac{1}{x - 6} - \frac{1}{x - 5} \\ \\ \frac{x - 2 - (x - 3)}{(x - 3)(x - 2)} = \frac{1}{x - 6} - \frac{1}{x - 5} \\ \\ faz \: o \: mmc \: de \: x - 6 \: e \: x - 5 \\ \\ \frac{x - 2 - (x - 3)}{(x - 3)(x - 2)} = \frac{(x - 5)}{(x - 6)(x - 5)} - \frac{(x - 6)}{(x - 5)(x - 6)} \\ \\ \frac{x - 2 - (x - 3)}{(x - 3)(x - 2)} = \frac{x - 5 - (x - 6}{(x - 6)(x - 5)} \\ \\ \frac{x - x - 2 + 3}{(x - 3)(x - 2)} = \frac{x - x - 5 + 6}{(x - 6)(x - 5)} \\ \\ \frac{1}{(x - 3)(x - 2)} = \frac{1}{(x - 6)(x - 5)} \\ \\ (x - 3)(x - 2) = (x - 6)(x - 5) \\ \\ (x - 3)(x - 2) - (x - 6)(x - 5) = 0 \\ \\ {x}^{2} - 2x - 3x + 6 - ({x}^{2} - 5x - 6x + 30 )= 0 \\ \\ {x}^{2} - 2x - 3x + 6 - ({x}^{2} - 11x + 30 )= 0 \\ \\ {x}^{2} - 2x - 3x + 6 -{x}^{2} + 11x - 30 = 0 \\ \\ organizando\\ \\ {x}^{2} - {x}^{2} - 2x - 3x + 11x - 30 + 6 \: = 0 \\ 0 - 5x + 11x - 24 = 0 \\ 6x = 24 \\ x = \frac{24}{6} \\ x = 4$

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