Spotting error in this solution: Solve for $x : [x] -2x= 4$

Question

Solution: Let $x \in [ k, k+1).$for some integer $ k$

Then $ [ x] = k$ where $[.]$ is the greatest integer function

Therefore, $ k -2x = 4$

$ \implies x = \dfrac{k-4}{2} \in [k,k+1)$

Solving for $k$, we get $k= -4,-5$ and therefore, $ x \in [-5,-3)$

But the points $ -5, -4.25$ clearly do not satisfy the equation.

Where is the error in this solution? Thanks!

Answer 1

You can set $x=k+r$ with $k\in\mathbb Z$ and $r\in[0,1)$

This is equivalent to your presentation, but easier to manipulate than keeping some $x$ and the equation becomes:

$\lfloor x\rfloor-2x=k-2(k+r)=-k-2r=4\quad$ therefore $2r$ is an integer

• $r=0\implies x=k=-4$
• $r=\frac 12\implies -k=4+2r=5$ and $x=-4.5$

Regarding your solution up to $k=-4,-5$ this seems correct. But how to you end up with $-5,-4.25$ after that ?

Reporting in $x=\frac{k-4}2$ you'll end up with $-4,-4.5$.