Is there a way to check inequalities in Mathematica given restrictions?

Question

Sorry if I'm still a noob, but is there a way to check inequalities in mathematica given some restrictions?

For example how could one go about checking whether:

Given $a,b,c\in\mathbb{R}$ and $a,b,c>0.5+\sqrt{5/4}$, is $abc(a+b+c)>3abc+ab+bc+ca$ true

in Mathematica

Answer 1

con = Thread[{a, b, c} > 1/2 + Sqrt[5/4]];

Simplify[Reduce[Flatten[
   {a b c (a + b + c) > 3 a b c + a b + b c + c a, con}],
  {a, b, c}], con]

True

Graphically,

Show[
 RegionPlot3D[
  a b c (a + b + c) > 3 a b c + a b + b c + c a,
  {a, 0, 3}, {b, 0, 3}, {c, 0, 3},
  PlotStyle -> Directive[Blue, Opacity[0.25]]],
 RegionPlot3D[And @@ con,
  {a, 0, 3}, {b, 0, 3}, {c, 0, 3},
  PlotStyle -> Red]]