# Isolate variable in inequality

I'm having the most stupid of problems. Suppose I have a simple inequality like $$|2-x|>2.$$

The solution to this is

$$x<0 , \,x>4.$$

This is pretty obvious, though Mathematica insists on spitting back the input:

Reduce[Abs[2 - x] > 2]


outputs

Abs[2 - x] > 2


How can I tell Mathematica I want a series of expression with only $$x$$ on the LHS? Trying to solve for $$x$$ only makes Mathematica complain that the solution is full-dimensional, I assume this means that Solve can only output single points and not intervals. I also suspect this has to do with the fact that Mathematica doesn't really know $$x$$ is real, but

FullSimplify[Reduce[Abs[2-x] > 2], Assumptions -> x \[Element] Reals]


produces the same result.

Try

Reduce[Abs[2 - x] > 2, x, Reals]
(*x < 0 || x > 4*)

• Well that was easy. Thanks, exactly what I needed. I'll accept when the 10 minutes pass. Commented Dec 8, 2020 at 12:57
• Thanks, you're welcome Commented Dec 8, 2020 at 14:46

I always avoid using Abs since it is a Complex Function. Here we use RealAbs

Reduce[RealAbs[2 - x] > 2]

• Also, Reduce[Sqrt[(2-x)^2] > 2, x] Commented Dec 8, 2020 at 15:58