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Q.
The number of solutions of $| [x] | - 2x | =4 $ where $[x]$ is the greatest integer is $\leq \, x$, is
Complex Numbers and Quadratic Equations
Solution:
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.