1. For which values of $a$ does the inequality $\left(a^{2}-1\right) x^{2}+2(a-1) x+2>0$ hold for every real number $x$?
Solution. Inequality (1), for $a^{2}-1 \neq 0$, holds for every real number $x$ if and only if the following conditions are satisfied:

$$
a^{2}-1>0, D=4\left(-a^{2}-2 a+3\right)0$, i.e., $a=1$ is also a solution.

Consequently, inequality (1) holds for every real number $x$ if and only if $a \in(-\infty,-3) \cup[1,+\infty)$.