Proof there is at least one root:
e^x = 3 - 2x
can you just do 0 = 3 - 2x
x = 1.5 ?
You can prove a solution exists without graphing, but you won't be able to find the exact value of the solution.
I won't go into detail here because the graphing should suffice. Basically note that e^x is always increasing (look at the derivative to see why) and 3 -2x is always decreasing. Since they both exist for all real x, and the range for y = 3 - 2x is all real values of y, they must intersect at some point. To write it out proper you may have to look into the intermediate value theorem.