6.179. $|x|+|x-1|=1$.

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.

6.179. $|x|+|x-1|=1$.
## Solution.

Let's plot the roots of the absolute values on the number line and consider the equation on the three intervals of the axis. We have three cases:

1) if $x<0$, then $-x-x+1=1$, from which we find $x=0$; there are no solutions, since $x<0$ does not belong to the considered interval;
2) if $0 \leq x<1$, then the equation takes the form $x-x+1=1$ or $1=1$; the solution will be $x \in[0 ; 1)$;

3) if $x \geq 1$, then we get $x+x-1=1$, from which $x=1$.

Answer: $x \in[0 ; 1]$.