BPT $\Leftrightarrow 2\sqrt[3]{x+2}+\sqrt[3]{3x+1}- \sqrt[3]{2x-1}\geq 0 (2)$
Đặt $f\left ( x \right )=2\sqrt[3]{x+2}+\sqrt[3]{3x+1}-\sqrt[3]{2x-1}$ BPT (2) có dạng $f\left ( x \right )\geq 0$
Xét $f\left ( x \right )=0\Leftrightarrow x=-2$
ta lại có $f\left ( -3 \right )=-2,0870..<0,f\left ( 1 \right )=3,471...>0$ nên $f\left ( x \right )\geq 0,\forall x\in \left[ {-2;+\infty } \right]$