Solve the inequality $$\frac{(x+1)(x-\sqrt3)}{x+2}

Question

Solve the inequality $$\frac{(x+1)(x-\sqrt3)}{x+2}<0,$$ giving your answer in exact form.[3] Hence solve, in exact form, $$\frac{(\tan\theta+1)(\tan\theta-\sqrt3)}{\tan\theta+2}<0,\qquad -\frac{\pi}{2}<\theta<\frac{\pi}{2}.$$[2]

Accepted Answer

(i) $x\in(-\infty,-2)\cup(-1,\sqrt3)$ (ii) $\theta\in\left(-\frac{\pi}{2},-\tan^{-1}2\right)\cup\left(-\frac{\pi}{4},\frac{\pi}{3}\right)$

Tangent substitution in a rational inequality

Tangent substitution in a rational inequality

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