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