I have the following equation as a question for homework: $\lfloor 2x \rfloor = 2\lfloor x\rfloor$
I know what the solution is by deducting to the possibilities. Meaning this equation is true for any x which is between n (an integer) and $n+y$ where $0\leq y < \frac{1}{2}$
putting it simple: \begin{align} x \in \lbrace n + y | n \in \mathbb{Z} , y \in [0, 0.5) \rbrace \end{align} I just don't know how to algebraically get to this solution. Help will be appreciated!
The best I can come up with is to replace manipulation of the floor function with manipulation of the fractional part function: $$\{x\} = x-\lfloor x\rfloor.$$
Then you have $\lfloor 2x\rfloor = 2\lfloor x \rfloor$ if and only if $2x- \{2x\} = 2(x - \{x\})$, which holds if and only if $2\{x\} = \{2x\}$, which holds if and only if $0\leq 2\{x\}\lt 1$, which holds if and only if $0\leq \{x\}\leq \frac{1}{2}$.
Would that do?