I agree with the post above that says this question is out of the scope of the GMAT, since you can't solve the resulting equation using conventional GMAT math, so I don't think people should worry about it much. But I suppose you can answer it without using any graphs.
If you think about the function f(x) = 2^x, that is a function that is 'constantly increasing'. If you plug in bigger and bigger values for x, the value of the function gets bigger and bigger. The same is true for the function h(x) = x, and if we add two functions that are constantly increasing, the result will be too. So the function g(x) = 2^x + x is a function that gets bigger and bigger as you plug in larger and larger values of x. Put algebraically,
if a > b, then g(a) > g(b)
and as a logical consequence, if we plug in two
different values for x into g(x), we can never get the same answer. So the equation g(x) = 2 can only have at most one solution for x.
Then we just need to confirm there is indeed one solution. This is where things really get outside the scope of the GMAT. If you notice that g(x) can be less than 2 (if you plug in x = 0, say) or greater than 2 (if you plug in x=1, say), then g(x) has to be equal to 2 for some value of x between 0 and 1, because g(x) is constantly and continuously increasing, and has to pass through the value 2 at some point. Technically I'm using something called the 'Intermediate Value Theorem' here though, which is a theorem from calculus that you don't ever need on the GMAT.
The logic above might not make a lot of sense to people who haven't studied calculus (you use this kind of reasoning in a lot of calc problems, but rarely in other kinds of problems), and as I pointed out above, if it doesn't make a lot of sense, it's not something you need to worry about if you're preparing for the GMAT!
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