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Question Number 133991 by EDWIN88 last updated on 26/Feb/21

$$\mid 3\mathrm{x}-\mid 4\mathrm{x}+2\mid \mid ⩾4>\mid 5\mathrm{x}+8\mid$$

Answered by benjo_mathlover last updated on 26/Feb/21

$$\mid 5\mathrm{x}+8\mid <4⩽\mid 3\mathrm{x}-\mid 4\mathrm{x}+2\mid \mid$$ $$(\bullet )\mid 5\mathrm{x}+8\mid <4\Rightarrow -4<5\mathrm{x}+8<4$$ $$\Rightarrow -12<5\mathrm{x}<-4;-\frac{12}{5}<\mathrm{x}<-\frac{4}{5}$$ $$(\bullet \bullet )\mid 3\mathrm{x}-\mid 4\mathrm{x}+2\mid \mid ⩾4$$ $$3\mathrm{x}-\mid 4\mathrm{x}+2\mid ⩽-4\cup 3\mathrm{x}-\mid 4\mathrm{x}+2\mid ⩾4$$ $$\mid 4\mathrm{x}+2\mid ⩾3\mathrm{x}+4\cup \mid 4\mathrm{x}+2\mid ⩽3\mathrm{x}-4$$ $$\left(\mathrm{i}\right)\mathrm{x}⩾-\frac{1}{2}\Rightarrow 4\mathrm{x}+2⩾3\mathrm{x}+4;\mathrm{x}⩾2$$ $$\mathrm{Solution}:\mathrm{x}⩾2$$ $$\left(\mathrm{ii}\right)\mathrm{x}⩽-\frac{1}{2}\Rightarrow -4\mathrm{x}-2⩾3\mathrm{x}+4;\mathrm{x}⩽-\frac{6}{7}$$ $$\mathrm{Solution}:\mathrm{x}⩽-\frac{6}{7}$$ $$\left(\mathrm{iii}\right)\mid 4\mathrm{x}+2\mid ⩽3\mathrm{x}-4$$ $$\mathrm{defined}\mathrm{for}3\mathrm{x}-4>0,\mathrm{x}>\frac{4}{3}$$ $$\mathrm{then}(7\mathrm{x}-2)(\mathrm{x}+6)⩽0$$ $$\mathrm{No}\mathrm{solution}.$$ $$\mathrm{Finally}\mathrm{solution}\mathrm{set}\mathrm{is}$$ $$\{-\frac{12}{5}<\mathrm{x}<-\frac{4}{5}\}\cap \{\mathrm{x}⩽-\frac{6}{7}\cup \mathrm{x}⩾2\}$$ $$\equiv -\underset{}{\stackrel{}{-}}<x⩽-\underset{}{\stackrel{}{-}}$$

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