Question

If $[x+\left[x\right]]\le 2$ where $\left[x\right]$ denotes the greatest integer $\le x,$ then $x$ lies in the interval,

• A. $(-\infty ,1)$
• B. $(-\infty ,2)$
• C. $(-\infty ,-2)$
• D. $(-\infty ,-1)$

Answer 1

The correct option is B $(-\infty ,2)$

$[x+\left[x\right]]\le 2$

$\Rightarrow x+\left[x\right]<3$

We know $x=\left[x\right]+\left\{x\right\}$, where $0\le \left\{x\right\}<1$

$\Rightarrow \left\{x\right\}+2\left[x\right]<3\Rightarrow 2\left[x\right]<3-\left\{x\right\}$

$\Rightarrow 2\left[x\right]<3\Rightarrow \left[x\right]<1.5\Rightarrow x<2$