## Task 5 - 120735

Determine all non-negative rational numbers $x$ that satisfy the equation $x+|x-1|=1$!
Let $x$ be any rational number for which $0 \leq x \leq 1$ holds. Then $x+|x-1|=x+1-x=1$, so the given equation is satisfied. Therefore, all rational numbers $x$ for which $x+|x-1|=x+1-x=1$ holds satisfy this equation.

Suppose there exists a rational number $x>1$ that satisfies the equation $x+|x-1|=1$. Then $1=x+|x-1|=x+x-1$ and thus $2x=2$, so $x=1$, which contradicts $x>1$.

Therefore, there is no rational number $x>1$ that satisfies the given equation. Hence, exactly those non-negative rational numbers $x$ for which $(0 \leq x \leq 1)$ holds satisfy the given equation.