Question

Решить неравенство √(x²+2x)> -3-x²

Answer 1

$\sqrt{x^2+2x}\ \textgreater \ -3-x^2; \\ \sqrt{f(x)}\ \textgreater \ g(x); \\ \left \ [ {{\left \{ {{g(x)\ \textless \ 0} \atop {f(x) \geq 0}} \right. } \atop {\left \{ {{g(x) \geq 0;} \atop {f(x)\ \textgreater \ g^2(x)}} \right. }} \right. \ \textless \ =\ \textgreater \ \left \ [ {{\left \{ {{-3-x^2\ \textless \ 0} \atop {x^2+2x \geq 0}} \right. } \atop {\left \{ {{-3-x^2 \geq 0;} \atop {x^2+2x)\ \textgreater \ (-3-x^2)^2}} \right. }} \right. \\ 1)-3-x^2\ \textless \ 0; \\ 3+x^2\ \textgreater \ 0; \\ x^2\ \textgreater \ -3; \\ x\in R. \\ 2)x^2+2x \geq 0; \\ x(x+2) \geq 0; \\ x\in (-\infty;-2]U[0;+\infty).$
$3) -3-x^2 \geq 0; \\ 3+x^2 \leq 0; \\ x^2 \leq -3; \\ x\in \emptyset; \\ 4)x^2+2x\ \textgreater \ (-3-x^2)^2; \\ x^2+2x-(-3-x^2)^2\ \textgreater \ 0; \\ x^2+2x-(9+6x^2+x^4)\ \textgreater \ 0; \\ x^2+2x-9-6x^2-x^4\ \textgreater \ 0; \\ -x^4-5x^2+2x-9\ \textgreater \ 0; \\ x^4+5x^2-2x+9\ \textless \ 0; \\ x\in \emptyset.$
ответ: $x\in (-\infty;-2]U[0:+ \infty).$