Q. If the solution set of the inequality $\log _x\left(\frac{5}{2}-\frac{1}{x}\right)>1$ is $(a, b) \cup(c, d)$ then find the value of $\frac{c d}{a b}$ where $(a < b < c < d).$
Complex Numbers and Quadratic Equations
Solution:
Domain: $x>\frac{2}{5} ; x \neq 1$
$\text { Case-I: } \text { If } x>1 \Rightarrow \frac{5}{2}-\frac{1}{x}>x \Rightarrow x+\frac{1}{x}<\frac{5}{2} $
$ 2\left(x^2+1\right)<5 x \Rightarrow 2 x^2-5 x+2<0$
$2 x^2-4 x-x+2<0$
