What are the values of $m$ for which inequality is true? $(m+2)e^(-2x)+2(m+2)e^-x+m>0$ .

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Answer 1 · 2017-04-26T22:38:43

$m > 2(1/(e^-x+1)^2-1)$

Calling $y = e^-x$ and assuming $m ne -2$ we have

$y^2+2y+m/(m+2) > 0$ or

$(y+1)^2-1 +m/(m+2) > 0$

or

$m/(m+2) > 1-(y+1)^2$

or

$m > 2(1-(y+1)^2)/(y+1)^2$

or

$m > 2(1/(y+1)^2-1)=2(1/(e^-x+1)^2-1)$

Attached a plot showing in light blue the feasible region in the plane $x xx m$

enter image source here

What are the values of mm for which inequality is true? (m+2)e^(-2x)+2(m+2)e^-x+m>0(m+2)e^(-2x)+2(m+2)e^-x+m>0.