2. Take $x \not= y$, but $f(x)=f(y)$, that is, you are assuming f is not 1-1. Then the inequality will not hold and the assumption (not 1-1) is false.
If the function is 1-1, then it will either be increasing, or decreasing. If mod is absolute value, then you will have cases: $f(a)>f(b), f(a)=f(b), f(a) Case 1 automatically proves it, case 2 can't happen from argument above. In case three, assume there is a c where f(c) is in the middle, and use the inequality to show that this can't happen.