Let #k = ln(5) =1.609438# and

let #m = ln(2) = 0.693147#

#5 = e^k##color(white)("XXXX")#and#color(white)("XXXX")##2 = e^m#

#5^(x+3) <= 2^(x+4)#

#color(white)("XXXX")#is equivalent to

#(e^k)^(x+3) <= (e^m)^(x+4)#

#kx+3k <= mx+4m#

#(k-m)x <= (4m-3k)#

#color(white)("XXXX")#since #k>m# we can divide both sides by #k-m#

#color(white)("XXXX")#without changing the orientation of the inequality

#x <= (4m-3k)/(k-m)#

Substituting the value of #ln(5)# for #k#

and the value of #ln(2)# for #m#

then pushing it all through my calculator...

I get #x <= -2.24353#