Question

Решить логарифмические неравенства.

Answer 1

$1)quad frac{log_2(1-x)}{x+1} textless 0; ,; ODZ:; ; left { {{1-x textgreater 0} atop {x+1ne 0}} right. ; ,; left { {{x textless 1} atop {xne -1}} right. \\ a); ; left { {{log_2(1-x) textless 0; ,; 1-x textgreater 0} atop {x+1 textgreater 0}} right. ; ; ili; ; b); ; left { {{log_2(1-x) textgreater 0; ,; 1-x textgreater 0} atop {x+1 textless 0}} right. \\a); ; left { {{1-x textless 1; ,; x textless 1} atop {x textgreater -1}} right. ; ,; left { {{x textgreater 0; ,; x textless 1} atop {x textgreater -1}} right. ; ; Rightarrow; ; 0 textless x textless 1$

$b); ; left { {{1-x textgreater 1; ,; x textless 1} atop {x textless -1}} right. ; ,; left { {{x textless 0; ,; x textless 1} atop {x textless -1}} right. ; ; Rightarrow ; ; ; x textless -1$

$Otvet:; ; xin (-infty textless -1)cup (0,1); .\\2)quad frac{1-log_{0,5}x}{sqrt{6x+2}} textless 0; ,; ; ODZ:; left { {{sqrt{6x+2}ne 0; ,; 6x+2 textgreater 0} atop {x textgreater 0}} right. ; left { {{x textgreater -frac{1}{3}} atop {x textgreater 0}} right. ; ,; ; x textgreater 0\\Tak; kak; ; sqrt{6x+2} textgreater 0,; to; drob textless 0,; esli; ; (1-log_{0,5}x) textless 0; .\\log_{0,5}x textgreater 1; ,; ; ; log_{0,5}x textgreater log_{0,5}0,5; ; Rightarrow ; ; x textless 0,5\\Otvet:; ; xin (0; ;; 0,5); .$

$3); ; ; Big |log_3(2-x)Big | textgreater 2; ,; ; ; ODZ:; ; 2-x textgreater 0; ; Rightarrow ; ; x textless 2\\star ; ; |x| textgreater a; ; ; Rightarrow ; ; ; underline {x textgreater a; ; ili; ; x textless -a}; ; , ; ; to; est; ; left [ {{x textgreater a} atop {x textless -a}} right. ; ; star \\ left [ {{log_3(2-x) textgreater 2} atop {log_3(2-x) textless -2}} right. ; ,; left [ {{2-x textgreater 3^2} atop {2-x textless 3^{-2}}} right. ; left [ {{x textless -7} atop {x textgreater frac{17}{9}}} right. \\ a); \; left [ {{x textless 2} atop {x textless -7}} right. ; ; ili; ; b); ; left [ {{x textless 2} atop {x textgreater frac{17}{9}}} right.$

$a); ; x textless -7; ; ; ili; ; ; b); ; frac{17}{9} textless x textless 2; ; ; ; (frac{17}{9}=1frac{8}{9})\\Otvet:; ; xin (-infty ,-7)cup (1frac{8}{9}, ,, 2); .$