Question

The number of solutions of $|[x]|-2x|=4$ where $[x]$ is the greatest integer is $\le x$, is

• A. $2$
• B. $4$
• C. $1$
• D. infinite

Answer 1

Answer: (B)

If $x=n\in Z,$ then $|n-2n|=4$
$\Rightarrow n=\pm 4$
If $x=n+K,n\in Z,0<K<1,$ then
$|n-2(n+K)|=4$
$\Rightarrow |-n-2K|=4$
It is possible if $K=\frac{1}{2}$. Then $|-n-1|=4$
$\Rightarrow n+1=\pm 4$
$\therefore n=3,-5$
$\therefore x$ has $4$ values.