2-x-2-6-

Question Number 159683 by SAMIRA last updated on 20/Nov/21

$$\mathrm{2}\:\leqslant\:\mid\boldsymbol{{x}}−\mathrm{2}\mid\:\leqslant\:\mathrm{6} \\ $$

Commented by tounghoungko last updated on 20/Nov/21

$$\:\mid{x}−\mathrm{2}\mid\geqslant\mathrm{2}\:\wedge\:\mid{x}−\mathrm{2}\mid\:\leqslant\:\mathrm{6} \\ $$$$\Rightarrow\left({x}−\mathrm{2}−\mathrm{2}\right)\left({x}−\mathrm{2}+\mathrm{2}\right)\geqslant\:\mathrm{0}\:\wedge\:\left({x}−\mathrm{2}−\mathrm{6}\right)\left({x}−\mathrm{2}+\mathrm{6}\right)\leqslant\:\mathrm{0} \\ $$$$\Rightarrow\left({x}−\mathrm{4}\right){x}\:\geqslant\:\mathrm{0}\:\wedge\:\left({x}−\mathrm{8}\right)\left({x}+\mathrm{4}\right)\leqslant\:\mathrm{0} \\ $$$$\Rightarrow\left[{x}\leqslant\:\mathrm{0}\:\cup\:{x}\geqslant\:\mathrm{4}\:\right]\:\wedge\:\left[−\mathrm{4}\:\leqslant\:{x}\:\leqslant\:\mathrm{8}\:\right] \\ $$$$\Rightarrow\:−\mathrm{4}\:\leqslant\:{x}\:\leqslant\:\mathrm{0}\:\cup\:\mathrm{4}\:\leqslant\:{x}\:\leqslant\:\mathrm{8}\: \\ $$

Answered by gsk2684 last updated on 20/Nov/21

$$\mathrm{2}\leqslant\mid{x}−\mathrm{2}\mid\leqslant\mathrm{6} \\ $$$$\mathrm{2}\leqslant\left({x}−\mathrm{2}\right)\leqslant\mathrm{6}\:{if}\:{x}\geqslant\mathrm{2}\: \\ $$$$\mathrm{4}\leqslant{x}\leqslant\mathrm{8} \\ $$$$ \\ $$$$\mathrm{2}\leqslant\left(\mathrm{2}−{x}\right)\leqslant\mathrm{6}\:{if}\:{x}\leqslant\mathrm{2}\: \\ $$$$\mathrm{0}\leqslant−{x}\leqslant\mathrm{4} \\ $$$$−\mathrm{4}\leqslant{x}\leqslant\mathrm{0} \\ $$$$ \\ $$

Answered by Acem last updated on 25/Oct/22

$$\:{x}\in\:\left[−\mathrm{4},\:\mathrm{0}\right]\:\cup\:\left[\mathrm{4},\:\mathrm{8}\right] \\ $$

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