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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.

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2 Answers 2

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Try

Reduce[Abs[2 - x] > 2, x, Reals]
(*x < 0 || x > 4*)
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  • $\begingroup$ Well that was easy. Thanks, exactly what I needed. I'll accept when the 10 minutes pass. $\endgroup$ Commented Dec 8, 2020 at 12:57
  • $\begingroup$ Thanks, you're welcome $\endgroup$ Commented Dec 8, 2020 at 14:46
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I always avoid using Abs since it is a Complex Function. Here we use RealAbs

Reduce[RealAbs[2 - x] > 2]
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    $\begingroup$ Also, Reduce[Sqrt[(2-x)^2] > 2, x] $\endgroup$
    – Bob Hanlon
    Commented Dec 8, 2020 at 15:58

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