Question

27^(x+1/3) - 10*9^(x) +10*3^(x) -5/9^(x+1/2) -10*3^(x) +3 ≤ 3^(x) +1/3^(x)-2 + 1/3^(x+1) -1

Answer 1

$3^x=a\\\\ \frac{3a^3-10a^2+10a-5}{3a^2-10a+3} \leq a+ \frac{1}{a-2} + \frac{1}{3a-1} \\ \frac{3a^3-10a^2+10a-5}{(3a-1)(a-3)} \leq \frac{a(3a-1)(a-2)+3a-1+a-2}{(3a-1)(a-2)} \\ \frac{(3a^3-10a^2+10a-5)(a-2)-(a-3)(a^2-2a)(3a-1)-(a-3)(4a-3)}{(3a-1)(a-3)(a-2)} \leq 0\\ \frac{3a^4-16a^3+30a^2-25a+10-3a^4+16a^3-23a^2+6a-4a^2+15a-9}{(3a-1)(a-3)(a-2)} \leq 0\\ \frac{3a^2-4a+1}{(3a-1)(a-3)(a-2)} \leq 0\\ \frac{(a-1)(3a-1)}{(3a-1)(a-3)(a-2)} \leq 0$

__-_(1/3)___-___[1]___+___(2)___-___(3)__+__

3ˣ = a ∈(0; 1/3) U (1/3; 1] U (2; 3)

x ∈ (-∞; -1) U (-1; 0] U (log_3(2); 1)