Practicing Success
The number of solutions of $|[x] -2x|= 4$, where [x] is the greatest integer less than or equal to x, is ______. |
4 |
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, $x = ± 4,\frac{7}{2},-\frac{9}{2}$ |