1.
(log3 x)^2 - 3log3 x + 2 > 0
Замена: log3 x = t
t^2 - 3t + 2 > 0
t^2 - 3t + 2 = 0
По теореме Виета:
t1 = 1
t2 = 2
t є (-оо; 1) U (2; +oo)
log3 x є (-оо; 1) U (2; +oo)
x є (0; 3) U (9; +oo)
2.
1/(3 - lg x) + 1/(1 + lg x) > 1
Замена: lg x = t
1/(3 - t) + 1/(1 + t) > 1
(1 + t + 3 - t - (3 - t)(1 + t)) / (3 - t)(1 + t) > 0
(4 - (3 + 3t - t - t^2)) / (3 - t)(1 + t) > 0
(t^2 - 2t + 1) / (3 - t)(1 + t) > 0
(t - 1)^2 / (3 - t)(1 + t) > 0
-1 1 3
—-o—-o—-o—-> t
(-) (+) (+) (-)
t є (-1; 1) U (1; 3)
lg x є (-1; 1) U (1; 3)
x є (0,1; 10) U (10; 1000)