Question is :

${\log }_{x^2}\left(\frac{x}{\left|x\right|}-x\right)⩾0$$\mathrm{Note} x\ne 0, \mathrm{as} \mathrm{the} \mathrm{base} \mathrm{is} x^2$$\mathrm{Case} \left(1\right) x>0$$\mathrm{so} \left|x\right|=x \mathrm{and} \frac{x}{\left|x\right|}=1$${\log }_{x^2}\left(1-x\right)⩾0, \mathrm{hence} 1-x⩾1 \mathrm{as} \mathrm{log1}=0$$-x⩾0, x⩽0$$\mathrm{but} x>0 \mathrm{and} x\ne 0$$\mathrm{So} \mathrm{no} \mathrm{solution} \mathrm{for} \mathrm{this} \mathrm{case}\left(1\right)$$\mathrm{Case} \left(2\right) x<0$$\mathrm{so} \left|x\right|=-x, \mathrm{and} \frac{x}{\left|x\right|}=-1$${\log }_{x^2}\left(-1-x\right)⩾0$$\mathrm{so} -1-x⩾1$$-x⩾2,$$x⩽-2$$\mathrm{Conclusion}: \mathrm{the} \mathrm{solution} \mathrm{set} \mathrm{is} x⩽-2$">