$$\left. \begin{array} { l } { ( a - 1 ) x ^ { 2 } - ( a + 1 ) x + ( a + 1 ) \gt 0 R } \end{array} \right.$$
$\left\{\begin{matrix}x\neq 2\text{, }&a=\frac{5}{3}\\x\in \mathrm{R}\text{, }&a>\frac{5}{3}\\x\in \left(-\infty,-\frac{\sqrt{-\left(3a-5\right)\left(a+1\right)}-a-1}{2\left(a-1\right)}\right)\cup \left(\frac{\sqrt{-\left(3a-5\right)\left(a+1\right)}+a+1}{2\left(a-1\right)},\infty\right)\text{, }&a>1\text{ and }a<\frac{5}{3}\\x<1\text{, }&a=1\\x\in \left(\frac{\sqrt{-\left(3a-5\right)\left(a+1\right)}+a+1}{2\left(a-1\right)},-\frac{\sqrt{-\left(3a-5\right)\left(a+1\right)}-a-1}{2\left(a-1\right)}\right)\text{, }&|a|<1\end{matrix}\right.$
$a>-\frac{1-x-x^{2}}{x^{2}-x+1}$
