Неравенство: x / (x^2+6x+8) >x / (x^2+7x+10)

1. Найдем корни трехчленов:

a) f (x) = x^2 + 6x + 8;

D/4 = 3^2 - 8 = 1; x = - 3 ± 1; x1 = - 4; x2 = - 2; f (x) = (x + 4) (x + 2).

b) g (x) = x^2 + 7x + 10;

D = 7^2 - 4 * 10 = 9; x = (-7 ± 3) / 2; x1 = - 5; x2 = - 2; g (x) = (x + 5) (x + 2).

2. Решим неравенство:

x / (x^2 + 6x + 8) > x / (x^2 + 7x + 10); x / (x + 4) (x + 2) - x / (x + 5) (x + 2) > 0; x / (x + 2) (1 / (x + 4) - 1 / (x + 5)) > 0; x / (x + 2) ((x + 5 - x - 4) / (x + 4) (x + 5)) > 0; x / (x + 2) (1 / (x + 4) (x + 5)) > 0; x / ((x + 2) (x + 4) (x + 5)) > 0; x ∈ (-∞; - 5) ∪ (-4; - 2) ∪ (0; ∞).

Ответ: (-∞; - 5) ∪ (-4; - 2) ∪ (0; ∞).