Решите логарифмическое неравенство

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3log(4) (x² + 5x + 6) ≤ 5 + log(4) (x + 2)³/(x + 3)

одз

(-3) (-2)

x∈ (-∞, -3) U (-2, +∞)

log(4) ((x + 2)(x + 3))³ ≤ log(4) 4^5 + log(4) (x + 2)³/(x + 3)

log(4) ((x + 2)(x + 3))³ ≤ log(4) 4^5*(x + 2)³/(x + 3)

снимаем логарифмы

(x + 2)³(x + 3)³ ≤ 4^5*(x + 2)³/(x + 3)

(x + 2)³(x + 3)³ - 4^5*(x + 2)³/(x + 3) ≤ 0

(x + 2)³((x + 3)³ - 4^5/(x + 3)) ≤ 0

(x + 2)³((x + 3)^4 - 1024)/(x + 3) ≤ 0

(x + 2)³((x + 3)² - 32)((x + 3)² + 32)/(x + 3) ≤ 0

(x + 3)² + 32 > 0 отбрасываем

(x + 2)³(x ² + 6х + 9 - 32)/(x + 3) ≤ 0

(x + 2)³(x ² + 6х  - 23)/(x + 3) ≤ 0

D = 36 + 4*23 = 128

x12 = (-6 +- √128)/2 = (-6 +- 4√8)/2 = -3 +- 2√8

x1 = -3 - 2√8 ≈ -8.65

x2 = -3 + 2√8 ≈ 3.65

(x + 2)³(x  + 3  + 2√8)(x + 3 - √8)/(x + 3) ≤ 0

Метод интервалов

[-3-2√8] (-3) [-2] [-3 + 2√8]

пересекаем с одз x∈ (-∞, -3) U (-2, +∞)

x ∈ [-3-2√8, -3) U (-2, -3+2√8]