All x satisfying the inequality (^-1x)^2-7(^-1x)+100, lie in the interval :

Question

All $x$ satisfying the inequality $(\cot^{-1}x)^2-7(\cot^{-1}x)+10>0$, lie in the interval :

Accepted Answer

$(\cot^{-1}x-5)(\cot^{-1}x-2)>0\Rightarrow\cot^{-1}x\in(-\infty,2)\cup(5,\infty)$.

Since $\cot^{-1}x\in(0,\pi)$ always, and $\pi<5$, the branch $(5,\infty)$ is impossible. So $\cot^{-1}x\in(0,2)$.

Since $\cot^{-1}$ is decreasing, this corresponds to $x\in(\cot2,\infty)$.

Solutions · 1