$\text{ The solution set of }{x}^2+2\le 3x\le 2x^2-5\text{ is }$

Solve the inequations

$\begin{array}{r}{x}^2-3x+2\le 0\text{ and }2x^2-3x-5\ge 0\\ \Rightarrow \left(x-1\right)\left(x-2\right)\le 0 and \left(2x-5\right)\left(x+1\right)\ge 0\\ \Rightarrow 1\le x\le 2\text{ and }x\le -1\text{ or }x\ge \frac{5}{2}\end{array}$

  The intersection of the two intervals is $\varphi$