решите неравенство 1 / (х-2) * (x-3) + 1 / (x-2) * (x-4) + 1/x^2-7x+12) <=1

Решим неравенство:

1 / ((х - 2) * (x - 3)) + 1 / ((x - 2) * (x - 4)) + 1 / (x^2 - 7 * x + 12) < = 1;

Найдем х в уравнении x^2 - 7 * x + 12 = 0;

D = (-7) ^2 - 4 * 1 * 12 = 49 - 48 = 1;

x1 = (7 + 1) / 2 = 8/2 = 4;

x2 = (7 - 1) / 2 = 6/2 = 3;

ОДЗ: х не равен 2, 3, 4.

Умножим неравенство на x^2 - 7 * x + 12.

1 / ((х - 2) * (x - 3)) * (x^2 - 7 * x + 12) + 1 / ((x - 2) * (x - 4)) * (x^2 - 7 * x + 12) + (x^2 - 7 * x + 12) * 1 / (x^2 - 7 * x + 12) < = 1 * (x - 3) * (x - 4);

1 / ((х - 2) * (x - 3)) * (x - 4) * (x - 3) + 1 / ((x - 2) * (x - 4)) * (x - 4) * (x - 3) + (x^2 - 7 * x + 12) * 1 / (x - 4) * (x - 3) < = 1 * (x - 4) * (x - 3);

(x - 4) / (x - 2) + (x - 3) / (x - 2) + 1 < = (x - 4) * (x - 3);

(x - 4) + (x - 3) + (x - 2) < = (x - 4) * (x - 3) * (x - 2);

x - 4 + x - 3 + x - 2 < = (x - 4) * (x - 3) * (x - 2);

3 * x - 9 < = (x - 4) * (x - 3) * (x - 2);

3 * (x - 3) - (x - 4) * (x - 3) * (x - 2) < = 0;

(x - 3) * (3 - (x - 4) * (x - 2)) < = 0;

(x - 3) * (3 - x^2 + 2 * x + 4 * x - 8) < = 0;

(x - 3) * (-x^2 + 6 * x - 5) < = 0;

(x - 3) * (x^2 - 6 * x + 5) > = 0;

(x - 3) * (x - 5) * (x + 1) > = 0;

{ x = 3;

x = 5;

x = - 1;

Тогда:

- + - +;

_ - 1 _ 3 _ 5 _;

Отсюда, - 1 < = x = 5.