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Given that $2[x]=x+2(x)$, $[x]$ if the Greatest Integer Function and $(x)$ is the fractional part of $x$, find the value (s) of $x$.

I tried replacing $(x)=x–[x]$ but for an equation in $x$ and $[x]$. How do I proceed???

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Let $x = n + r$ where $n \in \mathbb{Z}$ and $0 \le r \lt 1$. Then $[x] = n$ and $(x) = r$, so your equation becomes $2n = (n + r) + 2r \; \Rightarrow \; n = 3r$. Since $0 \le 3r \lt 3$, this gives $3$ choices for $n$ of $0, 1, 2$. You can then determine the matching values of $r$ and $x$.

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  • $\begingroup$ Thanks!!! Really helpful $\endgroup$ – user664431 Apr 16 at 17:37
  • $\begingroup$ @PranavGupta53535 You are welcome. Doing this type of substitution is the first thing I would try, and it often works to solve the problem $\endgroup$ – John Omielan Apr 16 at 17:41
  • $\begingroup$ Yeah... I forgot to use the inequality in (x") $\endgroup$ – user664431 Apr 16 at 17:46
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The equation you get by putting $(x)=x-[x]$ is $4[x]=3x$.

Now I believe the best approach would be to graph $[x]$ and $\frac34x$, both of which are pretty basic and easy to plot.

Search for the intersection points which are the required solutions.

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  • $\begingroup$ Thanks bro... Appreciate the help $\endgroup$ – user664431 Apr 16 at 17:23

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