The number of solutions of |[x] -2x|= 4, where [x] is the greatest in…

We have, $|[x]-2x|=4$

CASE I When $x ∈ Z$

In this case, we have [x] = x

$∴|[x] -2x|=4⇒x=4⇒ x=±4$

CASE II When $x ∉ Z$

In this case, we have

$x=n+λ$ where $n∈ Z$ and $0 <λ <1$

$⇒[x]=n$

$⇒|[x] -2x|=4$

$⇒|n−2 (n+λ)| = 4$

$⇒n+2λ=±4⇒n=±4-2λ$     ...(i)

This is possible, when $λ = \frac{1}{2}$

Putting $λ = \frac{1}{2}$ in (i), we get

$n = ±4-1$

$⇒n=3,-5⇒x=3+\frac{1}{2},-5+\frac{1}{2}⇒x=\frac{7}{2},-\frac{9}{2}$

Hence, $x = ± 4,\frac{7}{2},-\frac{9}{2}$