Question

If $[x+[2x\left]\right]<3,$ where $[.]$ denotes the greatest integer less than or equal to $x,$ then

• A. $x\in [0,1)$
• B. $x\in (-\infty ,\frac{2}{3}]$
• C. $x\in [0,\frac{3}{2})$
• D. $x\in (-\infty ,1)$

Answer 1

The correct option is D $x\in (-\infty ,1)$

$[x+[2x\left]\right]<3$
$\Rightarrow \left[x\right]+\left[2x\right]\le 2$
Any non-positive real number will satisfy this inequality.
For $x\in (0,\frac{1}{2})$
$\Rightarrow \left[x\right]=0,\left[2x\right]=0$
$\Rightarrow \left[x\right]+\left[2x\right]=0\le 2$
For $x\in (\frac{1}{2},1)$
$\left[x\right]=0,\left[2x\right]=1$
$\Rightarrow \left[x\right]+\left[2x\right]=1\le 2$
For $x\in (1,\frac{3}{2})$
$\left[x\right]=1,\left[2x\right]=2$
$\Rightarrow \left[x\right]+\left[2x\right]=3\nleqq 2$
$\therefore x\in (-\infty ,1)$