Question

How do you solve `absx>=x`?

Answer 1

With moduli, it is often helpful to split into cases at the point where the enclosed value changes sign.

Case (a): `x >= 0`
If `x >= 0` then `|x| = x >= x`.
So all `x >= 0` satisfy `|x| >= x`.

Case (b): `x < 0`
If `x < 0` then `|x| >= 0 > x`.
So all `x < 0` satisfy `|x| >= x`.

Combining the two cases we find `|x| >= x` is satisfied for all `x in RR`