Question

The number of integral solutions of the equation $\{x+1\}+2x=4[x+1]-6$, is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (B)

$1$

We know that, $\left\{x\right\}=x-\left[x\right]$
$\therefore \{x+1\}+2x=4[x+1]-6$
$\Rightarrow x+1-[x+1]+2x=4[x+1]-6$
$\Rightarrow 3x+1=5[x+1]-6$
$\Rightarrow 3x+1=5[x+1]-6$
$\Rightarrow 3x=5(\left[x\right]+1)-6$
$\Rightarrow 3x=5\left[x\right]-\mathrm{2...}\left(i\right)$
$\Rightarrow 3\{\left[x\right]+\left\{x\right\}\}=5\left[x\right]-2$
$\Rightarrow 3\left\{x\right\}=2\left[x\right]-2$
Now, $0\le \left\{x\right\}<1$
$\Rightarrow 0\le 3\left\{x\right\}<3$
$\Rightarrow 0\le 2\left[x\right]-2<3$
$\Rightarrow 1\le \left[x\right]<\frac{5}{2}$
$\Rightarrow \left[x\right]=1,2$
From Eq. (i) we have
$\left[x\right]=1\Rightarrow x=1$
$\left[x\right]=2\Rightarrow x=\frac{8}{3}$
Hence $x=1$ is the only integral solution.