# How do you solve absx>=x?

May 24, 2015

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

Case (a): $x \ge 0$
If $x \ge 0$ then $| x | = x \ge x$.
So all $x \ge 0$ satisfy $| x | \ge x$.

Case (b): $x < 0$
If $x < 0$ then $| x | \ge 0 > x$.
So all $x < 0$ satisfy $| x | \ge x$.

Combining the two cases we find $| x | \ge x$ is satisfied for all $x \in \mathbb{R}$