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Question Number 114593 by abdullahquwatan last updated on 19/Sep/20

((∣x−1∣)/(∣x∣−1)) ≤ 1

$$\frac{\mid{x}−\mathrm{1}\mid}{\mid{x}\mid−\mathrm{1}}\:\leqslant\:\mathrm{1} \\ $$

Answered by MJS_new last updated on 19/Sep/20

(1) ∣x∣−1≠0 ⇒ x≠±1  (2) ∣x∣−1<0 ⇒ −1<x<1         ∣x−1∣≥∣x∣−1 true ∀x∈R  ⇒ −1<x<1  (3) ∣x∣−1>0 ⇒ x<−1∨x>1         ∣x−1∣≤∣x∣−1 true for x≥1  ⇒ x>1    ⇒ −1<x<1∨x>1

$$\left(\mathrm{1}\right)\:\mid{x}\mid−\mathrm{1}\neq\mathrm{0}\:\Rightarrow\:{x}\neq\pm\mathrm{1} \\ $$$$\left(\mathrm{2}\right)\:\mid{x}\mid−\mathrm{1}<\mathrm{0}\:\Rightarrow\:−\mathrm{1}<{x}<\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\mid{x}−\mathrm{1}\mid\geqslant\mid{x}\mid−\mathrm{1}\:\mathrm{true}\:\forall{x}\in\mathbb{R} \\ $$$$\Rightarrow\:−\mathrm{1}<{x}<\mathrm{1} \\ $$$$\left(\mathrm{3}\right)\:\mid{x}\mid−\mathrm{1}>\mathrm{0}\:\Rightarrow\:{x}<−\mathrm{1}\vee{x}>\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\mid{x}−\mathrm{1}\mid\leqslant\mid{x}\mid−\mathrm{1}\:\mathrm{true}\:\mathrm{for}\:{x}\geqslant\mathrm{1} \\ $$$$\Rightarrow\:{x}>\mathrm{1} \\ $$$$ \\ $$$$\Rightarrow\:−\mathrm{1}<{x}<\mathrm{1}\vee{x}>\mathrm{1} \\ $$

Commented by abdullahquwatan last updated on 20/Sep/20

thanks sir

$${thanks}\:{sir} \\ $$

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