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Question Number 51721 by Tawa1 last updated on 29/Dec/18

$$\mathrm{Find}\mathrm{the}\mathrm{range}\mathrm{of}\mathrm{value}\mathrm{of}\mathrm{x}\mathrm{given}\mathrm{that}:\mid \mathrm{x}+\mathrm{i}\mid ⩾\frac{1}{\mathrm{x}}$$

Answered by ajfour last updated on 29/Dec/18

$$\sqrt{1+x^2}⩾\frac{1}{x}$$$$x<0andifx>0then$$$$let{x}^2=y$$$$\Rightarrow y(1+y)⩾1$$$$\Rightarrow y^2+y-1⩾0$$$${(y+\frac{1}{2})}^2⩾\frac{5}{4}$$$$\Rightarrow y⩾\frac{\sqrt{5}-1}{2}$$$$orx⩾\sqrt{\frac{\sqrt{5}-1}{2}}$$$$sox\in (-\mathrm{\infty },0)\cup [\sqrt{\frac{\sqrt{5}-1}{2}},\mathrm{\infty }).$$

Commented by Tawa1 last updated on 30/Dec/18

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

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