Решить неравенство [latex] \frac{1-3^{ x^{2} +2x-3}}{x^{2} +2x-3} \leq 0[/latex]

Решение смотри на фото.

[latex] \frac{1-3^{x^2+2x-3}}{x^2+2x-3} \leq 0 [/latex] let's [latex]t=x^2+2x-3[/latex] [latex] \frac{1-3^t}{t} \leq 0; [/latex] We have three possible cases: first: [latex] \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[/latex] [latex]t=x^2+2x-3\ \textgreater \ 0[/latex] [latex]x^2+3x-x-3\ \textgreater \ 0[/latex] [latex]x(x+3)-(x+3)\ \textgreater \ 0[/latex] [latex](x-1)(x+3)\ \textgreater \ 0[/latex] [latex][x-(1)]*[x-(-3)]\ \textgreater \ 0[/latex] [latex]x\in (-\infty;-3)\cup(1;+\infty)[/latex] second: [latex] \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[/latex] [latex]t=x^2+2x-3 \ \textless \ 0[/latex] [latex][x-(1)]*[x-(-3)]\ \textless \ 0[/latex] [latex]x\in (-3;1)[/latex] third: [latex] \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. [/latex] The system of inequalities behind have not sense due to its first inequality. ---------------------------------- So, we have: [latex]t\in (-\infty;0)\cup(0;+\infty)[/latex] and [latex]x^2+2x-3 \neq 0;[/latex] [latex](x-1)(x+3) \neq 0[/latex] [latex]x\in (-\infty;-3)\cup(-3;1)\cup(1;+\infty)[/latex] Answer: [latex](-\infty;-3)\cup(-3;1)\cup(1;+\infty)[/latex]