1-2e-x-1-gt-2-e-x-2-

Question Number 87550 by jagoll last updated on 05/Apr/20

\frac{1}{{2 e}^{- x} - 1} > \frac{2}{e^{- x} - 2}

Commented by john santu last updated on 05/Apr/20

\Rightarrow \frac{e^{x}}{2 - e^{x}} > \frac{{2 e}^{x}}{1 - {2 e}^{x}}

[let e^{x} = t]

\frac{t}{2 - t} > \frac{2 t}{1 - 2 t} \Rightarrow \frac{t - {2 t}^{2} - 4 t + {2 t}^{2}}{(2 - t) (1 - 2 t)} > 0

\frac{- 3 t}{(2 - t) (1 - 2 t)} > 0

\Rightarrow \frac{1}{2} < t < 2

e^{\ln (\frac{1}{2})} < e^{x} < e^{\ln (2)}

- \ln (2) < x < \ln (2)

Commented by jagoll last updated on 05/Apr/20

thank you sir

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