Решить неравенство [tex] \frac{1-3^{ x^{2} +2x-3}}{x^{2} +2x-3} \leq 0[/tex]. Created by asimx. algebra-ru

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$\frac{1-3^{x^2+2x-3}}{x^2+2x-3} \leq 0$
let's $t=x^2+2x-3$

$\frac{1-3^t}{t} \leq 0;$

We have three possible cases:

first:

$\left \{ {{\frac{1-3^t}{t} \leq 0|*t} \atop {t\ \textgreater \ 0}} \right. ; 
 \left \{ {{1-3^t \leq 0} \atop {t\ \textgreater \ 0}} \right. ; 
 \left \{ {{-3^t \leq -1} \atop {t\ \textgreater \ 0}} \right. ; 
 \left \{ {{3^t \geq 1} \atop {t\ \textgreater \ 0}} \right. ; 
 \left \{ {{3^t \geq 3^0} \atop {t\ \textgreater \ 0}} \right. ; 
 \left \{ {{t \geq 0} \atop {t\ \textgreater \ 0}} \right. ; 
t\ \textgreater \ 0$

$t=x^2+2x-3\ \textgreater \ 0$

$x^2+3x-x-3\ \textgreater \ 0$

$x(x+3)-(x+3)\ \textgreater \ 0$

$(x-1)(x+3)\ \textgreater \ 0$

$[x-(1)]*[x-(-3)]\ \textgreater \ 0$

$x\in (-\infty;-3)\cup(1;+\infty)$

second:

$\left \{ {{\frac{1-3^t}{t} \leq 0|*t} \atop {t\ \textless \ 0}} \right. ;
 \left \{ {{1-3^t \geq 0} \atop {t\ \textless \ 0}} \right. ;
 \left \{ {{-3^t \geq -1} \atop {t\ \textless \ 0}} \right. ;
 \left \{ {{3^t \leq 1} \atop {t\ \textless \ 0}} \right. ;
 \left \{ {{3^t \leq 3^0} \atop {t\ \textless \ 0}} \right. ;
 \left \{ {{t \leq 0} \atop {t\ \textless \ 0}} \right. ;
t\ \textless \ 0$

$t=x^2+2x-3 \ \textless \ 0$

$[x-(1)]*[x-(-3)]\ \textless \ 0$

$x\in (-3;1)$

third:

$\left \{ {{ \frac{1-3^t}{t} \leq 0} \atop {t=0}} \right. ;
 \left \{ {{ \frac{1-3^0}{0} \leq 0} \atop {t=0}} \right. ;
 \left \{ {{ \frac{0}{0} \leq 0} \atop {t=0}} \right.$

The system of inequalities behind have not sense due to its first inequality.
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So, we have: $t\in (-\infty;0)\cup(0;+\infty)$

and $x^2+2x-3 \neq 0;$

$(x-1)(x+3) \neq 0$

$x\in (-\infty;-3)\cup(-3;1)\cup(1;+\infty)$

Answer: $(-\infty;-3)\cup(-3;1)\cup(1;+\infty)$

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