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

Решим неравенство: 

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

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

(x 4) + (x 3) + (x 2) lt; = (x 4) * (x 3) * (x 2);

x 4 + x 3 + x 2 lt; = (x 4) * (x 3) * (x 2);

3 * x 9 lt; = (x 4) * (x 3) * (x 2);

3 * (x 3) (x 4) * (x 3) * (x 2) lt; = 0;

(x 3) * (3 (x 4) * (x 2)) lt; = 0;

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

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

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

(x 3) * (x 5) * (x + 1) gt; = 0;

x = 3;

x = 5;

x = -1;

Тогда:

 -     +    -    + ;

_ -1 _ 3 _ 5 _ ;   

Отсюда, -1 lt; = x lt; = 3 и x gt; = 5.