Find all $x$ for which $\left| x - \left| x-1 \right| \right| = \lfloor x \rfloor$

Question

Find all $x$ for which $$\left| x - \left| x-1 \right| \right| = \lfloor x \rfloor.$$

Any help would be appreciated. Thanks!

Answer 1

$\require{cancel}x=1$ can trivially be seen to work already.

if $x>1$:

$$|x-|x-1||=|x-(x-1)|=|1|=1=\lfloor x\rfloor\implies1<x<2$$

Assume $x\le1/2$:

$$|x-|x-1||=|x+(x-1)|=|2x-1|=2x-1=\lfloor x\rfloor\implies x=1/2,\xcancel1$$

Assume $1/2<x<1$:

$$|x-|x-1||=|x+(x-1)|=|2x-1|=1-2x=\lfloor x\rfloor\implies x=1/2$$

Solutions are $x=1/2,1\le x<2$

You can trivially check for the cases $x<1$ using the knowledge that $|2x-1|$ must be a whole number.

Answer 2

You can solve this by considering graphs of both sides, which will lead to the solution set $$x=\frac 12 \cup 1\leq x<2$$

The function on the left comprises a horizontal line $y=1$ for $x\geq 1$, and for $x\leq 1 $ you have a V shaped graph which is $y=|2x-1|$

You can see easily where this graph coincides with the floor function on the right