Absolute value inequality verification

Question

This is a pretty trivial question, but I'm trying to list out steps to show that if $|x-c|<1\Rightarrow |x|\leq |c|+1$. Is there a trick with the triangle inequality? Thanks

Answer 1

The triangle inequality has two versions

1) $|x-y|\leq |x|+|y|$

and

2)$||x|-|y||\leq |x-y|$

I think using number 2) you will manage something. Try!

Answer 2

$$|x|-|c|\leq |x-c| < 1$$ $$\Rightarrow |x|-|c| <1$$ $$\Rightarrow |x| <|c|+1$$