I am trying to solve the following problem: $|4-x| \leq |x|-2$. I am trying to do it algebraically, but I'm getting a solution to the problem that makes no sense. I fail to see the error in my reasoning though. I hope to get an explanation where I went wrong.
$|4-x| \leq |x|-2$
$|4-x|-|x| \leq -2$
If $4-x$ and $x$ are both negative, then for them to be equal to $2$, we need to multiply both expressions by $-1$.
$-4+x+x \leq -2$
$-4+2x \leq -2$
$2x \leq 2$
$x \leq 1$
But if you sub in any $x$ less than or equal to $1$, the inequality doesn't work! Can you please explain where in my logic, where in the steps, have I gone wrong? Thank you!