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Solve a complicated Inequation?

Posted 3 months ago
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Hello,

im having a problem to solve the following ineqtion for the variable $x$ per hand:

$$\left|\left(cx+x^2+1\right) \left(10c^2+11cx+3x^2-1\right) \right| =\left|\left(x\left(x+c\right)+1\right) \left(\left(2c+x\right)\left(5c+3x\right)-1\right)\right| <\left|\left(2c^2-3cx+x^2+1\right)^2\right|$$

with $x>0, -1\leq c <0$.

So i wanted to ask, if someone could help me and solve it with the help of mathematica.

It seems pretty straightforward:

Reduce[RealAbs[(c x + x^2 + 1) (10 c^2 + 11 c x + 3 x^2 - 1)] <
   RealAbs[(2 c^2 - 3 c x + x^2 + 1)^2] &&
  x > 0 && -1 <= c < 0,
 {c, x}]

The answer is in terms of Root objects. You can plot the set of solutions:

RegionPlot[
 RealAbs[(c x + x^2 + 1) (10 c^2 + 11 c x + 3 x^2 - 1)] < 
  RealAbs[(2 c^2 - 3 c x + x^2 + 1)^2], {c, -1, 0}, {x, 0, 10}]
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