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The solutions are

Spoiler

1 and 2.

 

Because
 

Spoiler

x 1/x = (x2)1/x^2 = x 2/x^2

By inspection 1 and 2 are solutions, and a plot of the difference of x 1/x and x 2/x^x shows sign changes (only) at 1 and 2.

Also, for any value of x not equal to 0 (illegal division) or 1 (all powers of 1 are equal) dividing by x yields simply

1/x = 2/x2 or more simply 

x2 = 2x.

Cute.

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