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Question Number 87536 by M±th+et£s last updated on 04/Apr/20

$$solve$$$$\mid 2x-1\mid =3\lfloor x\rfloor +2\left\{x\right\}$$$$$$

Answered by mr W last updated on 05/Apr/20

$$x=n+\delta$$$$\lfloor x\rfloor =n$$$$\left\{x\right\}=\delta$$$$2x-1=2n+2\delta -1$$$$$$$$if2x-1⩾0,i.e.x⩾\frac{1}{2}:$$$$2n+2\delta -1=3n+2\delta$$$$\Rightarrow 2n-1=3n\Rightarrow n=-1$$$$\Rightarrow \delta =anyvalue0⩽\delta <0$$$$\Rightarrow -1⩽x<0\Rightarrow nosolution$$$$$$$$if2x-1<0,i.e.x<\frac{1}{2}:$$$$-2n-2\delta +1=3n+2\delta$$$$5n+4\delta =1$$$$4\delta mustbeintger.$$$$\delta =0:5n=1\Rightarrow nosolution$$$$\delta =\frac{1}{4}:5n=0\Rightarrow n=0\Rightarrow x=\frac{1}{4}<\frac{1}{2}\Rightarrow solution$$$$\delta =\frac{1}{2}:5n=-1\Rightarrow nosolution$$$$\delta =\frac{3}{4}:5n=-2\Rightarrow nosolution$$$$$$$$\Rightarrow onlysolutionisx=\frac{1}{4}.$$

Commented by M±th+et£s last updated on 05/Apr/20

$$thankyousirnicesolution$$

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