For [tex]x\ \textgreater \ 0[/tex], [tex]2^x\ \textless \ 3^x[/tex], all the more [tex]2^x\ \textless \ 3^x+1[/tex], therefore no solution.
For [tex]x=0[/tex], we have [tex]2^0=3^0+1 \Rightarrow 1=1+1 \Rightarrow 1=2[/tex]. Obviously that' false.
For [tex]x\ \textless \ 0[/tex] both [tex]2^x[/tex] and [tex]3^x[/tex], where [tex]2^x\ \textgreater \ 3^x[/tex], belong to the range [tex](0,1)[/tex], then [tex]3^x+1[/tex] belong to the range [tex](1,2)[/tex].
Those ranges don't have a common part, therefore, again, no solution.
So, the equation [tex]2^x=3^x+1 [/tex] doesn't have a solution.
