# How can I solve this inequality for every $n$ in Mathematica?

I am trying to find the solution to the inequality $$t^{-1/n}<1$$ when $$n$$ is an integer and $$t$$ is positive, however when using

Reduce[t^(-1/n) >= 1, t]


it returned an error message stating that reduce was not able to solve this inequality (Mathematica 12.0). I also tried

Assuming[Element[n, Integers] && n > 0, Reduce[t^(-n^(-1)) <= 1, t]]


If I use

f[n_] := Reduce[t^(-1/n) <= 1, t];
f /@ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}


I get the correct intervals of $$t\geq1$$. How can I show this for every $$t>0$$ and $$n\geq1$$ using Mathematica.

• wolfram alpha can solve this. – Xminer Nov 19 '19 at 2:47
• Mathematica is not yet capable of reading minds, so you need to explicitly tell it your assumptions: FullSimplify[Reduce[t^(-1/n) < 1 && t > 1 && n ∈ Integers && n > 0, t], t > 1 && n ∈ Integers && n > 0] – J. M.'s torpor Nov 19 '19 at 2:52
• But you are basically implementing the answer and checking whether it is true. What if the answer was not available or not immediate? That is why I posed the question. – DMH16 Nov 19 '19 at 2:58

## 1 Answer

You state that n is an integer and that t is positive. Include both of these constraints in Reduce

Reduce[t^(-1/n) <= 1 && Element[n, Integers] && t > 0, t] // Simplify

(* t^(-1/n) ∈ Reals &&
n ∈ Integers && ((n <= -1 && 0 < t <= 1) || (n >= 1 && t >= 1)) *)