Solve the inequality $- \frac{1}{2} < \frac{x^{3}}{1 + x^{6}}$ $-1/2<x^3/(1+x^6)$ ?

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Solve the inequality $- \frac{1}{2} < \frac{x^{3}}{1 + x^{6}}$ $-1/2<x^3/(1+x^6)$ ?

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Answer 1 · 2018-06-04T19:09:02

${(x^{3} + 1)}^{2} > 0$ $(x^3+1)^2>0$ and $x \neq - 1$ $xne-1$

Explanation:

Multiplying by $1 + x^{6} > 0$ $1+x^6>0$
we get
$- 1 - x^{6} < 2 x^{3}$ $-1-x^6<2x^3$
so we have
$0 < x^{6} + 2 x^{3} + 1$ $0<x^6+2x^3+1$
this is
${(x^{3} + 1)}^{2} > 0$ $(x^3+1)^2>0$ for $x \neq - 1$ $xne-1$

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