Question

The number of solution(s) of the equation $\left[x\right]+2\{-x\}=3x$, is$/$are (where $\left[\right]$ represents the greatest integer function and $\left\{x\right\}$ denotes the fractional part of x$)$:

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is A $1$

Given; $\left[x\right]+2\{-x\}=3x$

$\{-x\}+[-x]=-x(\because \left\{a\right\}+[a]=a)$

$\therefore \{-x\}=-x-[-x]$

We know that, $[-x]=-1-\left[x\right]$

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

$\left[x\right]-2x+2+2\left[x\right]=3x$

$3\left[x\right]=5x-2$

L.H.S is integer

$\therefore$ R.H.S must be integer

$\therefore$ $x$must be integer

As $x$ is integer , $\left[x\right]=x$

$3x=5x-2$

$x=1$

Only one solution is possible.