Hello!

I have solved this system of equations (see below) in Mathematica for $x$ where the coefficients of the equations/inequalities are functions of $a,b$ and $c$. Mathematica then displays real solutions $x$ with constraints on $a,\,b$ and $c$.

The constraints for $c$ (for example) are written in function of roots objects $\mbox{Root}[,k]$. In the output, I see for instance $\mbox{Root}[,1] < c \leq \mbox{Root}[,2]$. On the other hand, I also see the condition $0<c<\mbox{Root}[,3]$.

I assume that $\mbox{Root}[,1] < \mbox{Root}[,2]$? However, can I also assume that $\mbox{Root}[,2] < \mbox{Root}[,3]$? Moreover, given that Mathematica displays the output like this, I can assume these roots are all real? Otherwise, it would be quite meaningless?

The Mathematica code for the system is:

Reduce[
     16 x^4 - 40 a x^3 + (15 a^2 + 24 b) x^2 - 18 a b x + 3 b^2 == 0 
  && 5 a x - 4 x^2 - b > 0 
  && 15 a x - 20 x^2 - 3 b < 0 
  && 4 x^3 - 8 c x^2 + 5 c a x - c b > 0 
  && c > 0 && x > 0, 
     x, Reals]

I also asked the question somewhere else but it seems that most people do not really prefer these Root Objects. Indeed, it's much easier to work with analytical expression, but since I know the numerical values of $(a,b,c)$ these roots can be computed quite easily with a numerical solver. I just need to be sure that I interpret them correctly.

Many thanks! Cheers