<s>[INST] How do I solve the inequality $x^{12}−x^9+x^4−x+1>0$ using intervals? My book has the question $x^{12}−x^{9}+x^{4}−x+1>0$. The solution given gives three cases, when $x \le 0$, when $0 < x \le 1$ and when $x > 1$. How did they get these intervals? What is the method used?
 [/INST] Hint: When $x=0$ we get $1>0$ and when $x<0$ then $-x^9,-x>0$ so $$x^{12}-x^9+x^4-x+1>0$$
For $$0<x\le 1$$ we get $$x=1$$ then we get $$1>0$$ and we substitute $$x=\frac{1}{y}$$ so we obtain
$$1+y^3(y^5-1)+y^{11}(y-1)>0$$
and for $x>1$ we have $$x^9(x^3-1)+x(x^3-1)+1>0$$
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