Question

How many real numbers satisfy the relation $\left[x\right]=\frac{3}{2}x$.

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Answer 1

We can write the given relation as $\left\{x\right\}=\frac{2}{3}\left[x\right]$
We know $0\le \left\{x\right\}<1$.
$\Rightarrow 0\le \frac{2}{3}\left[x\right]<1$
$\Rightarrow 0\le \left[x\right]<\frac{3}{2}$

So the values [x] can assume are 0 and 1. For any other value of [x],

For $\left\{x\right\}=\frac{2}{3}\left[x\right]$

When $\left[x\right]=0\Rightarrow x=0$

Similarly when $\left[x\right]=1\Rightarrow \left\{x\right\}=\frac{2}{3}$
$x=\left[x\right]+x$
$\Rightarrow x=\frac{5}{3}$
So the two values of x are $0$ and $\frac{5}{3}$