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Question Number 193079 by Mastermind last updated on 03/Jun/23

$$\mathrm{Find}\mathrm{all}\mathrm{x}\in \mathbb{R}\mathrm{that}\mathrm{satisfy}\mathrm{the}\mathrm{following}$$$$\mathrm{inequalities}:$$$$\mathrm{a})\mid 4\mathrm{x}-3\mid ⩽11$$$$\mathrm{b})\mid \mathrm{x}-2\mid >\mid \mathrm{x}+1\mid$$$$\mathrm{c})\mid \mathrm{x}\mid +\mid \mathrm{x}+2\mid +\mid 2-\mathrm{x}\mid ⩽8$$

Commented by MM42 last updated on 04/Jun/23

$$a)-11⩽4x-3⩽11\Rightarrow -8⩽4x⩽14\Rightarrow -2⩽x⩽3.5$$$$b)ifx⩽-1\Rightarrow -x+2>-x-1\Rightarrow 2>-1isalwayscorrect\Rightarrow x⩽-1\left(i\right)$$$$if-1⩽x⩽2\Rightarrow -x+2>x+1\Rightarrow x<0.5\Rightarrow -1⩽x<0.5\left(ii\right)$$$$if2⩽x\Rightarrow x-2>x+1\Rightarrow -2>1\Rightarrow isincorrect$$$$\Rightarrow ans=\left(i\right)\cup \left(ii\right)=(-\mathrm{\infty },0.5)$$$$c)x⩽-2\Rightarrow -x-x-2+2-x⩽8\Rightarrow x⩾-\frac{8}{3}\Rightarrow -\frac{8}{3}⩽x⩽-2\left(i\right)$$$$-2⩽x⩽0\Rightarrow -x+x+2+2-x⩽8\Rightarrow x⩾-4\Rightarrow -2⩽x⩽0\left(ii\right)$$$$0⩽x⩽2\Rightarrow x+x+2+2-x⩽8\Rightarrow x⩽4\Rightarrow 0⩽x⩽2\left(iii\right)$$$$2⩽x\Rightarrow x+x+2-2+x⩽8\Rightarrow x⩽\frac{8}{3}\Rightarrow 2⩽x⩽\frac{8}{3}\left(iv\right)$$$$ans=\left(i\right)\cup \left(ii\right)\cup \left(iii\right)\cup \left(iv\right)=[-\frac{8}{3},\frac{8}{3}]$$$$$$

Commented by Subhi last updated on 03/Jun/23

$$\left(C\right)\mid a+b+c\mid ⩽\mid a\mid +\mid b\mid +\mid c\mid$$$$\mid x\mid +\mid x+2\mid +\mid x-2\mid ⩾\mid x+x+x+2-2\mid =\mid 3x\mid$$$$\downarrow$$$$\mid 2-x\mid$$$$\mid 3x\mid ⩽8$$$$\frac{-8}{3}⩽x⩽\frac{8}{3}$$

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