Question

Как решить это неравенство? Можете подробно расписать решение?

Answer 1

2/(5^x - 1) + (5^x - 2)/(5^x - 3) ≥ 2

5^x ≠ 1  x≠0

5^x - 3 ≠ 0   x ≠ log(5) 3

(5^x - 2)/(5^x - 3) = (5^x - 3 + 1)/(5^x - 3) = 1 + 1/(5^x - 3)

2/(5^x - 1) +  1 + 1/(5^x - 3) ≥ 2

2/(5^x - 1) +   1/(5^x - 3) - 1 ≥ 0

5^x = t  > 0

2/(t - 1) + 1/(t - 3) - 1 ≥ 0

(2(t - 3) + (t - 1) - (t - 1)(t - 3))/(t - 1)(t - 3) ≥ 0

(2t - 6 + t - 1 - t² + 4t - 3)/(t - 1)(t - 3) ≥ 0

(-t² + 7t - 10)/(t - 1)(t - 3) ≥ 0

(t - 2)(t - 5)/(t - 1)(t - 3) ≤ 0  

++++++(1) --------------- [2] +++++++++ (3) ------------[5] +++++++++

t ∈ (1, 2] U (3, 5]

5^x = t

5^x > 1

x > 0

5^x ≤ 2

x ≤ log(5) 2

5^x > 3

x > log(5) 3

5^x ≤5

x ≤ 1

ответ x ∈ (0, log(5) 2] U (log(5) 3, 1]