What are possible values of x if # lne^(2x)/2<xln(3)?#?

Question asked for 'values' so you have to state all of them.
#x in ]0,oo]# Notation for the range of zero to infinity but excluding 0

Explanation:

#ln(e^(2x)) # has the same value as #2xln(e)#
#ln(e) = 1# so we now have:

#(2x)/2 < x ln(3)#

#x < x ln(3)#

#0 < xln(3) - x#

#0< x(ln(3) -1)#

#0/(ln(3) -1) < x#

# x > 0#

This inequality is true for any positive real number #x in (0;+oo)#

Explanation:

#ln(e^(2x))/2 < xln(3)#

#(2xlne)/2 < xln3#

#x < xln3#

#x(1-ln3) < 0#

#x>0#

I changed the sign of inequality in last expression because #ln3~~1.098#, so last step was the division by a negative number (#1-ln3#)