> I know lots of other equations, simpler ones for that matter, > that have no solution (in R, say) for which no corresponding > thing can be done, e.g., x+5=x-4.
And I know of one who would have a solution for your equation: Wallis Budge! :-)
C. P. Snow [1, p. 9] wrote this story:
<quote> One of the classical examples is that of Wallis Budge, who later became a world famous Egyptologist, being coached for a university entrance examination. He was coached by E. W. Hobson, Hardy's predecessor at Cambridge, and Hobson used to tell the story in a most lugubrious tone:
I said to Budge, "Budge, if 2X = 1, what does X equal?" and Budge would turn his great wise eyes at me, and think, and think, and think, and say "-2." </quote>
1. C. P. Snow: Mathematicians and Dunderheads. The Griffin, vol. 9, no. 10, New York: October 1960, pp. 3 - 9
PS: The British writer Sir Charles P. Snow is known to mathematicians as a friend of Hardy. (He wrote a Foreword in Hardy's _Apology_)