Răspuns:
[tex][x] \in \mathbb Z[/tex], deci [tex]|x-1|=[x]-sgn(x-1)\in\mathbb Z[/tex]. Rezulta ca [tex]x\in\mathbb Z[/tex].
Pe de alta parte, [tex]|x-1|\geq 0[/tex] deci [tex] [x]-sgn(x-1)=x-sgn(x-1)\geq 0[/tex]. Cum [tex]sgn(x-1)\geq -1[/tex], rezulta [tex]x+1\geq 0[/tex], deci [tex]x\geq -1[/tex].
Daca x=-1, atunci [tex]|x-1|=|-2|=2\neq -1+1=0 [/tex], fals.
Daca x=0, atunci [tex]|x-1|=|-1|=1 = 0 - sgn(-1)=1[/tex], adevarat.
Daca x=1, atunci [tex]|x-1|=0=1-sgn(0)=1[/tex], fals.
Daca [tex]x\geq 2[/tex], atunci [tex]|x-1|=x-1=x-sgn(x-1)=x-1[/tex] adevarat.
In concluzie, multimea solutiilor ecua\c tiei este [tex]\{0,2,3,4,5,\cdots\}[/tex].
