1 / (x-5) - 1 / (x-7) - 1 / (x1) + 1 / (x-3) = 0

1. Сгруппируем дроби и приведем к сумме двух дробей:

1 / (x-5) - 1 / (x-7) - 1 / (x - 1) + 1 / (x - 3) = 0;

[1 / (x-5) - 1 / (x-7) ] + [1 / (x - 3) - 1 / (x - 1) ] = 0;

[1 / (x-5) * (x-5) (x-7) / (x-5) (x-7) - 1 / (x-7) * (x-5) (x-7) / (x-5) (x-7) ] + [1 / (x - 3) * (x - 3) (x - 1) / (x - 3) (x - 1) - 1 / (x - 1) * (x - 3) (x - 1) / (x - 3) (x - 1) ] = 0;

[ (х - 7) - (х - 5) / (x-5) (x-7) ] + [ (х - 1) - (х - 3) / (x - 3) (x - 1) ] = 0;

[ (х - 7 - х + 5) / (x-5) (x-7) ] + [ (х - 1 - х + 3) / (x - 3) (x - 1) ] = 0;

[ - 2 / (x-5) (x-7) ] + [2 / (x - 3) (x - 1) ] = 0;

2. Приведем дробь к одному общему знаменателю:

[ - 2 / (x-5) (x-7) * (x-5) (x-7) (x - 3) (x - 1) / (x-5) (x-7) (x - 3) (x - 1) ] + [2 / (x - 3) (x - 1) * (x-5) (x-7) (x - 3) (x - 1) / (x-5) (x-7) (x - 3) (x - 1) ] = 0;

[ - 2 (x - 3) (x - 1) + 2 (x-5) (x-7) ] / (x-5) (x-7) (x - 3) (x - 1) = 0;

3. Дробь равна нулю, если числитель равен нулю:

- 2 (x - 3) (x - 1) + 2 (x-5) (x-7) = 0;

2 ((x-5) (x-7) - (x - 3) (x - 1)) = 0;

(x-5) (x-7) - (x - 3) (x - 1) = 0;

x² - 7x - 5x + 35 - (x² - x - 3x + 3) = 0;

x² - 7x - 5x + 35 - x² + x + 3x - 3) = 0;

- 8x + 32 = 0;

- 8 х = - 32;

х = 4;

Ответ: х = 4.