show equation has one solution

here is a question our prof gave us:

Show that the equation X + e^(X) - 5 = 0 has only one root in the interval [1,2].

i graphed the equation and saw that it is between 1 and 2

but i cant calculate it for sum reason. any ideas how? can u solve it using regular algebra? i dont thing wer allowed using calc but if thers no other way, i mite jus use calc if thers a way lol.

thanks.

Define
Use the fact that this function is continuous over a connected space, so you have the intermediate value theorem. Check that and gives you that there exists some for which . To show this point is unique, it suffices to show this function is strictly monotonic on this interval. How do you do this? I don't know if you have had calculus or not (if so, just take the derivative to see it is always positive: is always positive), it is pretty clear this function is increasing everywhere by inspection since both and have this property and the sum of strictly monotonic functions must also be strictly monotonic. The -5 term at the end doesnt affect anything, why? by definition, for f to be strictly monotonic you need only show see what happens to the constant term?.