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Question Number 86128 by M±th+et£s last updated on 27/Mar/20

$$\lfloor 2x-\frac{1}{2}\rfloor =\lfloor \mid x\mid -\frac{1}{2}\rfloor =2x-2$$

Answered by mr W last updated on 27/Mar/20

$$\lfloor 2x-\frac{1}{2}\rfloor -\lfloor \mid x\mid -\frac{1}{2}\rfloor =2x-2$$$$\lfloor 2x-\frac{1}{2}\rfloor -\lfloor \mid x\mid -\frac{1}{2}\rfloor =2x-2=n=\{\begin{array}{l}2k\\ 2k+1\end{array}$$$$\Rightarrow x=\frac{n}{2}+1=\{\begin{array}{l}k+1\\ k+\frac{3}{2}\end{array}$$$$casen⩾-2ork⩾-1$$$$x⩾0$$$$\lfloor 2x-\frac{1}{2}\rfloor =\lfloor n+1+\frac{1}{2}\rfloor =n+1=\{\begin{array}{l}2k+1\\ 2k+2\end{array}$$$$\lfloor \mid x\mid -\frac{1}{2}\rfloor =\lfloor \frac{n}{2}+\frac{1}{2}\rfloor =\{\begin{array}{l}k\\ k+1\end{array}$$$$2k+1-k=2k$$$$\Rightarrow k=1\Rightarrow n=2\Rightarrow x=2$$$$2k+2-k-1=2k+1$$$$\Rightarrow k=0\Rightarrow n=1\Rightarrow x=\frac{3}{2}$$$$$$$$casen<-2ork<-1$$$$x<0$$$$\lfloor 2x-\frac{1}{2}\rfloor =\lfloor n+1+\frac{1}{2}\rfloor =n+1=\{\begin{array}{l}2k+1\\ 2k+2\end{array}$$$$\lfloor \mid x\mid -\frac{1}{2}\rfloor =\lfloor -\frac{n}{2}-2+\frac{1}{2}\rfloor =\{\begin{array}{l}-k-2\\ -k-2\end{array}$$$$2k+1+k+2=2k$$$$\Rightarrow k=-3\Rightarrow n=-6\Rightarrow x=-2$$$$2k+2+k+2=2k+1$$$$\Rightarrow k=-3\Rightarrow n=-5\Rightarrow x=-\frac{3}{2}$$$$$$$$summaryofsolutions:$$$$x=-2,-\frac{3}{2},\frac{3}{2},2$$

Commented by M±th+et£s last updated on 27/Mar/20

$$godblessyousir.butithinkthatx=-\frac{3}{2}isasolutiontoo$$

Commented by mr W last updated on 27/Mar/20

$$yes.ioverlookedavalue.$$

Commented by M±th+et£s last updated on 27/Mar/20

$$nicesolutionsirthankyou$$

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