Let's denote the number of cans already on the shelf as \( x \).
We know from the problem that the shelf can hold a total of 58 cans, and Alex can still put 39 cans on the shelf before it's full. This means that the number of cans currently on the shelf plus the number of cans Alex can still add should equal the total capacity of the shelf.
We can set up the equation based on this information:
\[ x + 39 = 58 \]
Now, we can solve for \( x \):
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Subtract 39 from both sides: \[ x = 58 - 39 \]
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Perform the subtraction: \[ x = 19 \]
Thus, the number of cans already on the shelf is \( \boxed{19} \).