Question

Решить неравенство. Подробно!

Answer 1

y=log4(x)

(y+3)/(y-3)+(y-3)/(y+3)>=(4y+16)/(y^2-9)

((y+3)^2+(y-3)^2-4y-16)/((y+3)(y-3))>=0

(y-1)^2/((y+3)(y-3))>=0

Метод интервалов 

y ∈ (-oo;-3)U{1}U{3;+oo)

x ∈ (0;1/64)U{4}U{64;+oo)

Answer 2

ОДЗ  x>0⇒x∈(0;∞)
(log(4)x+3)/(log(4)x-3)+(log(4)x-3)/(log(4)x+3)-
-(4log(4)x+16)/[(log(4)x-3)(log(4)x+3)]≥0
log(4)x=a
((a+3)/(a-3)+(a-3)/(a+3)-(4a+16)/[(a-3)(a+3)]≥0
[(a+3)²+(a-3)²-(4a+16)]/([(a-3)(a+3)]≥0
(a²+6a+9+a²-6a+9-4a-16)/[(a-3)(a+3)]≥0
(2a²-4a+2)/[(a-3)(a+3)]≥0
2(a²-2a+1)/[(a-3)(a+3)]≥0
2(a-1)²/[(a-3)(a+3)]≥0
a-1=0⇒a=1
a-3=0⇒a=3
a+3=0⇒a=-3 
                  +                      _                      _                     +
----------------------(-3)-----------------[1]-------------------(3)---------------------
a<-3⇒log(4)x<-3⇒x<1/64 +ОДЗ⇒0<x<1/64
a=1⇒log(4)x=1⇒x=4
a>3⇒log(4)x>3⇒>x>64
Ответ x∈(0;1/64) U (64;∞) U {4}