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Question Number 155393 by cortano last updated on 30/Sep/21

  ∣∣x−4∣−4∣ ≥ 4

$$\:\:\mid\mid\mathrm{x}−\mathrm{4}\mid−\mathrm{4}\mid\:\geqslant\:\mathrm{4}\: \\ $$

Answered by mr W last updated on 30/Sep/21

∣x−4∣−4≥4   ⇒∣x−4∣≥8   ⇒x−4≥8 or x−4≤−8   ⇒x≥12 or x≤−4  or  ∣x−4∣−4≤−4   ⇒∣x−4∣≤0   ⇒x=4    solution:  x≤−4, x=4 or x≥12

$$\mid{x}−\mathrm{4}\mid−\mathrm{4}\geqslant\mathrm{4}\: \\ $$$$\Rightarrow\mid{x}−\mathrm{4}\mid\geqslant\mathrm{8}\: \\ $$$$\Rightarrow{x}−\mathrm{4}\geqslant\mathrm{8}\:{or}\:{x}−\mathrm{4}\leqslant−\mathrm{8}\: \\ $$$$\Rightarrow{x}\geqslant\mathrm{12}\:{or}\:{x}\leqslant−\mathrm{4} \\ $$$${or} \\ $$$$\mid{x}−\mathrm{4}\mid−\mathrm{4}\leqslant−\mathrm{4}\: \\ $$$$\Rightarrow\mid{x}−\mathrm{4}\mid\leqslant\mathrm{0}\: \\ $$$$\Rightarrow{x}=\mathrm{4} \\ $$$$ \\ $$$${solution}: \\ $$$${x}\leqslant−\mathrm{4},\:{x}=\mathrm{4}\:{or}\:{x}\geqslant\mathrm{12} \\ $$

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