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Answer: $7$


$3\oplus 5 = 3\times 5 + 3 +5 = 23$  and  $2\oplus x = 2x + 2+ x= 3x + 2$.
 
These are equal, so $3x + 2 = 23$, i.e. $x=7$.
 
 
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.