Question

The set of values of $x$ for which the inequality ${\left[x\right]}^2-5\left[x\right]+6\le 0$ (where $[.]$ denote the greatest integral function) holds good are

• A. $2\le \left[x\right]<3$
• B. $2\le x<4$
• C. $2\le \left[x\right]\le 3$
• D. $\left(b\right)$ and $\left(c\right)$ both

Answer 1

The correct option is D $\left(b\right)$ and $\left(c\right)$ both

We have, ${\left[x\right]}^2-5\left[x\right]+6\le 0$

Let $\left[x\right]=y$

$\Rightarrow y^2-5y+6\le 0$

$\Rightarrow (y-2)(y-3)\le 0$

$\Rightarrow y\in [2,3]$

$\Rightarrow \left[x\right]\in [2,3]$

$\therefore x\in [2,4)$