Question

Which of the following is a solution of $\left(\log x\right)(\frac{5}{2}-\frac{1}{x})>(\frac{5}{2}-\frac{1}{x})?$

• A. $(\frac{1}{3},\frac{2}{5})$
• B. $(1,2)$
• C. $(\frac{2}{5},\frac{1}{2})$
• D. None of these

Answer 1

The correct option is A $(\frac{1}{3},\frac{2}{5})$

$\left(\log x\right)(\frac{5}{2}-\frac{1}{x})>(\frac{5}{2}-\frac{1}{x})$

$\Rightarrow (\frac{5}{2}-\frac{1}{x})(\log x-1)>0$

$\Rightarrow \left(\frac{5x-2}{2x}\right).(\log x-1)>0$

Case-I: $\frac{5x-2}{2x}>0$ and $(\log x-1)>0$

$\Rightarrow xϵ(-\infty ,0)\cup (\frac{2}{5},\infty )$ and $x>10$

$\therefore xϵ(10,\infty )$

Case-II: $\frac{5x-2}{2x}<0$ and $\left(\log x\right)-1<0$

$\Rightarrow xϵ(0,\frac{2}{5})$ and $0<x<10$

$\therefore xϵ(0,\frac{2}{5})$

Hence, $(\frac{1}{3},\frac{2}{5})$ is a solution of the inequality given.