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Question Number 170868 by Tawa11 last updated on 01/Jun/22

$$\mathrm{Solve}:\mid \mathrm{x}-1\mid +\mid \mathrm{x}-2\mid ⩾4$$

Answered by thfchristopher last updated on 02/Jun/22

$$\Rightarrow \mid x-1\mid ⩾4-\mid x-2\mid$$$$\Rightarrow x-1⩾4-\mid x-2\mid \mathrm{or}x-1⩽\mid x-2\mid -4$$$$\mathrm{Case}1:$$$$x-1⩾4-\mid x-2\mid$$$$\Rightarrow \mid x-2\mid ⩾5-x$$$$\Rightarrow x-2⩾5-x\mathrm{or}x-2⩽x-5\left(\mathrm{rejected}\right)$$$$\Rightarrow 2x⩾7$$$$\Rightarrow x⩾\frac{7}{2}$$$$\mathrm{Case}2:$$$$x-1⩽\mid x-2\mid -4$$$$\Rightarrow \mid x-2\mid ⩾x+3$$$$\Rightarrow x-2⩾x+3\left(\mathrm{rejected}\right)\mathrm{or}x-2⩽-x-3$$$$\Rightarrow 2x⩽-1$$$$\Rightarrow x⩽-\frac{1}{2}$$$$\mathrm{Conclusion}:x⩽-\frac{1}{2}\mathrm{or}x⩾\frac{7}{2}$$

Commented by Tawa11 last updated on 02/Jun/22

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

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