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

Question

Find all $x$ for which $$\left| x - \left| x-1 \right| \right| = \lfloor x \rfloor.$$Express your answer in interval notation.

I started by looking at x < 1 and x $\geq$ 1 separately, but I got stuck on what to do next.

Answer 1

Suppose the equation holds. Then $\lfloor x \rfloor \geq 0$ since it is equal to an absolute value. Consequently, we have $x \geq 0$.

Case 1: $0 \leq x < 1$. Then we have $x - (1 - x) = 0$. Then $x = \frac{1}{2}$.

Case 2: $x \geq 1$. Then we have $x - (x - 1) = 1 = \lfloor x \rfloor$. This can only be true if $x \in [1, 2)$.

Then if $x$ solves the equation, we have either $x = \frac{1}{2}$ or $x \in [1, 2)$.

It's easy to check that $x = \frac{1}{2}$ or $x \in [1, 2)$ implies that $x$ is a solution to the equation. Then the solution set is

$\{\frac{1}{2}\} \cup [1, 2) = [\frac{1}{2}, \frac{1}{2}] \cup [1, 2)$

Answer 2

We need to consider the following cases

1) $x\ge 1$

$$\left| x - \left| x-1 \right| \right| =1= \lfloor x \rfloor$$

2) $x< 1$

$$\left| 2x-1\right| = \lfloor x \rfloor$$

and then

2a) $\frac12\le x< 1$

$$ 2x-1 = \lfloor x \rfloor$$

2b) $x< \frac12$

$$ 1-2x = \lfloor x \rfloor$$