$|x-1| +|x+1| <1$
Can you help me with this inequality?
$|x-1| +|x+1| <1$.
I think that it doesn't exist an $x$ that satisfies this inequality, but I'm not really convinced about my answer.
Solution 1:
Note that $|y|\geq y$ implies that for all $x\in\mathbb{R}$, $$| x-1| +| x+1|=| 1-x| +|x+1|\geq (1-x)+(x+1)\geq 2.$$ Hence $| 1-x| +|x+1|<1$ is never satisfied.
Solution 2:
Use $|a|=a$, if $a\ge 0$ and $|a|=-a$ if $a<0.$
1) If $x<-1$, then $|x-1|+|x+1|=1-x-x-1<1$ $$-2x<1$$ $$x>-\frac12$$ But $-1>x$
2) $-1\le x \le 1$
Similarly
3) $x>1$
Similarly