Question

Find all values of $a$ for which the inequality $(a-1)x^2-(a+1)x+a+1>0$ is satisfied for all real $x$.

Answer 1

$(a-1)x^2-(a+1)x+a+1>0$

For this inequality to hold

$(a+1{)}^2-4(a+1\left)\right(a-1)<0$

$(a+1)(a+1-4a+4)<0$

$(a+1)(3a-5)>0⟹a\in (-\infty ,-1)\cup (\frac{5}{3},\infty )$