Question

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

• A. $2\le \left[x\right]<3$
• B. $2\le x<4$
• C. $2\le x<3$
• D. $2\le x\le 4$

Answer 1

Answer: (B)

$2\le x<4$

Given that

${\left[x\right]}^2-5\left[x\right]+6\le 0\Rightarrow (\left[x\right]-2)(\left[x\right]-3)\le 0\Rightarrow 2\le \left[x\right]\le 3$

$\Rightarrow \left[x\right]=2or3$

$\Rightarrow \left[x\right]=2$ $\left[x\right]=3$

$\Rightarrow 2\le x<3$ $3\le x<4$ $\dots \left[\right[t]=x\Rightarrow x\le t<x+1]$

from the above inequalities

$\therefore 2\le x<4$