It is equavalent $\{x^2\}\ge 2\{x\}$. It has more intervals in solution. $[1,\sqrt 2),[2,\sqrt 5),[1+\sqrt 2,\sqrt 6),[3,\sqrt{10}),[1+\sqrt 5,\sqrt{11}),[1+\sqrt 6,\sqrt{12}),[4,\sqrt{17}),[1+\sqrt{10},\sqrt{18}),[1+\sqrt{11},\sqrt{19}),[1+\sqrt{12},\sqrt{20})$.