# Solving four inequalities in three unknowns

I would like to solve the following set of inequalities in Mathematica.

$$m \geq 1$$

$$n \geq 1$$

$$d > 5$$

$$d - \frac{m}{2}(d-2) - n(d-1) > 0$$

I get no solutions by hand. I'd like to check this answer using Mathematica.

How do I do this in Mathemtica?

• Why do you need Mathematica for this? Obviously, the second and third term are negative and smallest (in magnitude) for m=1 and n=1. Even then the result is -1/2 d + 2 which is <0. – Felix Feb 24 '17 at 1:39
• Reduce[{m >= 1, n >= 1, d > 5, d - m/2 (d - 2) - n (d - 1) > 0}] returns False indicating that the conditions are inconsistent. – Bob Hanlon Feb 24 '17 at 6:36
• @ felix . I have it by hand, but I want to check my answer using all possible means. – nightmarish Feb 24 '17 at 11:19

Reduce gives False instantaneously and FindInstance gives an empty list. So your surmise of no solution seems to be correct.

Reduce[{d - m/2 (d - 2) - n (d - 1) > 0, m >= 1, n >= 1,
d > 5}, {m, n, d}]

(* False *)

FindInstance[{d - m/2 (d - 2) - n (d - 1) > 0, m >= 1, n >= 1,
d > 5}, {m, n, d}]

(* {} *)


@Felix 's response is best but if you had to do it with Mathematica:

Maximize[{m + n - d (m/2 + n), m >= 1 && n >= 1 && d > 5}, {m, n, d}]


Maximize::wksol: Warning: there is no maximum in the region in which the objective function is defined and the constraints are satisfied; a result on the boundary will be returned.

{-(11/2), {m -> 1, n -> 1, d -> 5}}