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Question Number 98806 by bemath last updated on 16/Jun/20

$$\mathrm{for}a\mathrm{is}\mathrm{integer}\mathrm{number}\mathrm{such}\mathrm{that}$$$$\mid \mid x-1\mid -2\mid ⩽a\mathrm{exactly}\mathrm{has}2013$$$$\mathrm{solution}$$

Commented by Rasheed.Sindhi last updated on 16/Jun/20

$$Assumingxaninteger:$$$$\mid \mid x-1\mid -2\mid ⩽a$$$$\Rightarrow \{\begin{array}{l}\mid x-1-2\mid ⩽a\Rightarrow \mid x-3\mid ⩽a...\left(i\right)\\ \mid -x+1-2\mid ⩽a\Rightarrow \mid -x-1\mid ⩽a..\left(ii\right)\end{array}$$$$\left(i\right)\Rightarrow \{\begin{array}{l}x-3⩽a\Rightarrow x⩽a+3\\ -x+3⩽a\Rightarrow x⩾-a+3\end{array}..\left(iii\right)$$$$\left(ii\right)\Rightarrow \{\begin{array}{l}-x-1⩽a\Rightarrow x⩾-a-1\\ x+1⩽a\Rightarrow x⩽a-1\end{array}..\left(iv\right)$$$$\left(iii\right)\Rightarrow -a+3⩽x⩽a+3......\left(v\right)$$$$\left(iv\right)\Rightarrow -a-1⩽x⩽a-1......\left(vi\right)$$$$\left(v\right)\mathrm{\&}\left(vi\right)\Rightarrow -a-1⩽x⩽a+3$$$$-1004-1⩽x⩽1004+3$$$$-1005⩽x⩽1007$$$$x=-1005,-1004,...,-2,-1,0,1,2...,1006,1007$$$$a=1004$$

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