$|x - 1| = 1 - x$

Question

$$|1 - x| = 1 - x$$ $$\iff -(x - 1) = 1 - x \text{ or } x - 1 = 1 -x$$ $$\iff -x + 1 = 1 - x \text{ or } 2x = 2$$ $$\iff 0 = 0 \text{ or } x = 1$$

The $0 = 0$ is bothering me. This answer does not make any sense. What mistake did I do here?

Answer 1

When you simplify equations, you're really relating propositions. For example, $x+1=y+1$ is a proposition about those two variables.

When we "subtract 1 from both sides to get $x=y$, we're creating a new proposition, and by the rules of algebra, we know that the new one is true if and only if the old one is.

If, by a sequence of such propositions that are logically equal, you end up at one that is always true, like $0=0$, that makes every one in the chain true. If you end up with $1=0$, then it makes the whole chain false.

That example is simple. Sometimes, one proposition turns into a compound one: $x^2=1$ turns into $x=1$ OR $x=-1$.

In the case of absolute value, you have to be pretty careful. The proposition $|x|=a$ is true if and only if $(x<0\: AND\: -x=a) OR (x\ge 0\: AND\: x=a)$. That is four propositions in one.

I think the comments have already pointed out where exactly you went wrong. I hope this answer helps clarify what it means to get to 0=0.