Решить неравенство 1/x^2+8x-9 > = 1/3x^2-5x+2

1. Найдем корни трехчленов:

1 / (x^2 + 8x - 9) ≥ 1 / (3x^2 - 5x + 2);

1) f (x) = x^2 + 8x - 9;

D/4 = 4^2 + 9 = 25 = 5^2; x = - 4 ± 5; x1 = - 9; x2 = 1; f (x) = (x + 9) (x - 1).

2) g (x) = 3x^2 - 5x + 2;

D = 5^2 - 4 * 3 * 2 = 1; x = (5 ± 1) / 6; x1 = 4/6 = 2/3; x2 = 1; g (x) = 3 (x - 2/3) (x - 1) = (3x - 2) (x - 1).

2. Приведем к общему знаменателю:

1 / (x + 9) (x - 1) - 1 / (3x - 2) (x - 1) ≥ 0; ((3x - 2) - (x + 9)) / (x + 9) (x - 1) (3x - 2) ≥ 0; (3x - 2 - x - 9) / (x + 9) (x - 1) (3x - 2) ≥ 0; (2x - 11) / (x + 9) (x - 1) (3x - 2) ≥ 0.

3. Корни двучленов:

x1 = - 9; x2 = 2/3; x3 = 1; x4 = 11/2.

4. Решение неравенства:

x ∈ (-∞; - 9) ∪ (2/3; 1) ∪ [11/2; ∞).

Ответ: (-∞; - 9) ∪ (2/3; 1) ∪ [11/2; ∞).