Question

Решите неравенство:

Answer 1

$log_{3}(x^2-9)-3log_{3} frac{x+3}{x-3} textgreater 2$
ОДЗ:
$left { {{x^2-9 textgreater 0} atop { frac{x+3}{x-3} textgreater 0 }} right.$
$(x-3)(x+3) textgreater 0$

---------+-------(-3)--------- - --------(3)--------+--------
/////////////////////                           //////////////////////
$x$ ∈ $(-$ ∞ $;-3)$ ∪ $(3;+$ ∞ $)$

$log_{3}[(x-3)(x+3)]-3(log_{3} (x+3)-log_3(}{x-3})) textgreater 2$
$log_{3}(x-3)+log_3(x+3)-3log_{3} (x+3)+3log_3(}{x-3}) textgreater 2$
$4log_{3}(x-3)-2log_{3} (x+3) textgreater 2$
$2log_{3}(x-3)-log_{3} (x+3) textgreater 1$
$2log_{3}(x-3) textgreater log_{3} (x+3)+ 1$
$log_{3}(x-3)^2 textgreater log_{3} (x+3)+ log_33$
$log_{3}(x-3)^2 textgreater log_{3}[ (x+3)*3]$
$log_{3}(x-3)^2 textgreater log_{3} (3x+9)$
$(x-3)^2 textgreater 3x+9$
$x^{2} -6x+9-3x-9 textgreater 0$
$x^{2} -9x textgreater 0$
$x(x-9) textgreater 0$
------+------(0)------ - -------(9)-------+----------
/////////////////                     /////////////////////
учтем ОДЗ

Ответ: (- ∞; -3) ∪ (9; + ∞)