Question

If $\left[x\right]$ denotes the greatest integer function, then the solution set of the inequation $\frac{\left[x\right]-1}{2-\left[x\right]}>0,$ is

• A. $R-\left\{2\right\}$
• B. $(1,2)$
• C. $\varphi$
• D. $R-\{1,2\}$

Answer 1

The correct option is C $\varphi$

$\frac{\left[x\right]-1}{2-\left[x\right]}>0\Rightarrow \frac{\left[x\right]-1}{\left[x\right]-2}<0$
By wavy curve method,
$\left[x\right]\in (1,2)$
Since, $\left[x\right]$ is always an integer,
$x\in \varphi$