{x}-{2x}=x
{x}-{2x}=x
Rezolvati in $\mathbb{R}$ ecuatia $\left\lbrace x \right\rbrace-\left\lbrace2x\right\rbrace=x.$
Liceul Teoretic Cobani
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dangerous storm
- Mesaje: 145
- Membru din: Joi Iul 03, 2014 9:29 pm
Re: {x}-{2x}=x
Ecuatia data este echivalenta cu $x+\{x\}=2x+\{2x\}$,adica $[x]+2\{x\}=[2x]+2\{2x\}$,ceea ce implica $2\{2x\}-2\{x\}\in\mathbb{Z}$.Cum $1>\{2x\}-\{x\}>-1$,rezulta ca $2>2\{2x\}-2\{x\}>-2$,deci $2\{2x\}-2\{x\}\in \{1,0-1\}$, ceea ce este echivalent cu $\{2x\}-\{x\}\in \{\frac{1}{2},0,\frac{-1}{2}\}$.
Daca $\{2x\}-\{x\}=\frac{1}{2}$,atunci $x=\{x\}-\{2x\}=\frac{-1}{2}$.Insa $\{\frac{-1}{2}\}-\{2\cdot\frac{-1}{2}\}=\frac{1}{2}\neq \frac{-1}{2}$.
Daca $\{2x\}-\{x\}=0$,atunci $x=\{x\}-\{2x\}=0$,iar $\{0\}-\{2\cdot 0\}=0$.
Daca $\{2x\}-\{x\}=\frac{-1}{2}$,atunci $x=\{x\}-\{2x\}=\frac{1}{2}$,iar $\{\frac{1}{2}\}-\{2\cdot \frac{1}{2}\}=\frac{1}{2}$.
In concluzie,$x\in\{0,\frac{1}{2}\}$.
Daca $\{2x\}-\{x\}=\frac{1}{2}$,atunci $x=\{x\}-\{2x\}=\frac{-1}{2}$.Insa $\{\frac{-1}{2}\}-\{2\cdot\frac{-1}{2}\}=\frac{1}{2}\neq \frac{-1}{2}$.
Daca $\{2x\}-\{x\}=0$,atunci $x=\{x\}-\{2x\}=0$,iar $\{0\}-\{2\cdot 0\}=0$.
Daca $\{2x\}-\{x\}=\frac{-1}{2}$,atunci $x=\{x\}-\{2x\}=\frac{1}{2}$,iar $\{\frac{1}{2}\}-\{2\cdot \frac{1}{2}\}=\frac{1}{2}$.
In concluzie,$x\in\{0,\frac{1}{2}\}$.