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I am a beginner and cannot read the result given by the reduce command kindly help. I used:

Reduce[(8 x^3 - x^3 (1 + x^2)^(3/2))/(8 x^3 + (1 + x^2)^(3/2) (-5 + 4 x^3))>0]

My result :

-Sqrt[3] < x < 0 ||Root[25 + 75 #1^2 - 40 #1^3 + 75 #1^4 - 120 #1^5 - 23 #1^6 - 120 #1^7 + 48 #1^8 - 40 #1^9 + 48 #1^10 + 16 #1^12 &, 1] < x < Sqrt[3]

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  • $\begingroup$ Welcome to Mathematica.SE! To allow people to give you an answer that you will find useful, please explain in detail which part of the result you don't understand. (E.g. operators like ||? The Root object?) $\endgroup$ – Szabolcs Sep 1 '18 at 9:49
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There are no closed-form expressions for the roots of a generic polynomial of degree greater then 4. Root[...,1] is Mathematica's way to symbolically express the first root. By converting to machine precision numbers, you can obtain something that is easier to interpret:

N[
 Reduce[(8 x^3 - 
      x^3 (1 + x^2)^(3/2))/(8 x^3 + (1 + x^2)^(3/2) (-5 + 4 x^3)) > 0,
   x]
 ]

-1.73205 < x < 0. || 0.877633 < x < 1.73205

Here, || has to be read as logical "or". So the solution set is approximately

$$\{x \in \mathbb{R} | -1.73205 < x < 0 \text{ or } 0.877633 < x < 1.73205 \}$$ which is equal to $${]-1.73205,0[} \cup {]0.877633,1.73205[}.$$

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