The common solution set of the inequations $x^2-4x \leq 12$ and $x^2-2x \geq 15$ taken together is

The solution set contained in $R^{+}$ of the inequation $3^x + 3^{1-x} - 4 < 0$ is

The solution set of $x^{2} \leq 4$ is......

For the inequality $2^{\log_{\sqrt{2}}(x - 1)} > x + 5$,the set of real values of $x$ is:

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2 - (t^2 - 5t + 6)x + 1 = 0$,where $t \in \mathbb{R}$,and $a_n = \alpha^n + \beta^n$. Then the minimum value of $\frac{a_{2023} + a_{2025}}{a_{2024}}$ is

Let $S$ be the set of positive integral values of $a$ for which $\frac{ax^2+2(a+1)x+9a+4}{x^2-8x+32} < 0, \forall x \in R$. Then,the number of elements in $S$ is: