# Greatest Integer Function linear equation

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???

## 2 Answers

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 $$x$$.

• Thanks!!! Really helpful – user664431 Apr 16 at 17:37
• @PranavGupta53535 You are welcome. Doing this type of substitution is the first thing I would try, and it often works to solve the problem – John Omielan Apr 16 at 17:41
• Yeah... I forgot to use the inequality in (x") – user664431 Apr 16 at 17:46

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.

• Thanks bro... Appreciate the help – user664431 Apr 16 at 17:23