Solve for x
\left\{\begin{matrix}x\neq -\frac{1}{2}\text{, }&a=\frac{3\sqrt{2}}{10}\\x\in \left(-\infty,-\frac{\sqrt{10a^{2}+2\sqrt{2}a-3}+\sqrt{2}a+1}{2\left(\sqrt{2}a+1\right)}\right)\cup \left(\frac{\sqrt{2}\left(\sqrt{2\left(10a^{2}+2\sqrt{2}a-3\right)}-2a-\sqrt{2}\right)}{4\left(\sqrt{2}a+1\right)},\infty\right)\text{, }&a>\frac{3\sqrt{2}}{10}\\x\in \left(\frac{\sqrt{2}\left(\sqrt{2\left(10a^{2}+2\sqrt{2}a-3\right)}-2a-\sqrt{2}\right)}{4\left(\sqrt{2}a+1\right)},-\frac{\sqrt{10a^{2}+2\sqrt{2}a-3}+\sqrt{2}a+1}{2\left(\sqrt{2}a+1\right)}\right)\text{, }&a<-\frac{\sqrt{2}}{2}\\x\in \mathrm{R}\text{, }&a\geq -\frac{\sqrt{2}}{2}\text{ and }a<\frac{3\sqrt{2}}{10}\end{matrix}\right.
Solve for a
\left\{\begin{matrix}a>-\frac{\sqrt{2}\left(x^{2}+x+1\right)}{2\left(x^{2}+x-1\right)}\text{, }&x<\frac{-\sqrt{5}-1}{2}\text{ or }x>\frac{\sqrt{5}-1}{2}\\a\in \mathrm{R}\text{, }&x=\frac{\sqrt{5}-1}{2}\text{ or }x=\frac{-\sqrt{5}-1}{2}\\a<-\frac{\sqrt{2}\left(x^{2}+x+1\right)}{2\left(x^{2}+x-1\right)}\text{, }&x>\frac{-\sqrt{5}-1}{2}\text{ and }x<\frac{\sqrt{5}-1}{2}\end{matrix}\right.
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