There are only two possibilities for the multiplication
We cannot use a 1 since that would make the product be the
same as the other number.  Those two possibilities 
for the multiplication are:

2×3 = 6  and 2×4 = 8

Let's try 2×4 = 8

That leaves 1,3,5,6,7,9

The addition could be 1+5=6 leaving 3,5, and 7. 
But no subtraction can be made from those.

The addition could be 1+6=7 leaving 3,5, and 9. 
But no subtraction can be made from those.

The addition could be 3+6=9 leaving 1,5, and 7. 
But no subtraction can be made from those.

So we have ruled out 2×4=8 for the multiplication

So the multiplication can only be 2×3 = 6 

That leaves 1,4,5,7,8,9

The addition could be 1+4=5 leaving 7, 8, and 9. 
But no subtraction can be made from those.


The addition could be 1+8=9 leaving 4, 5, and 7. 
But no subtraction can be made from those.

The addition could be 1+7=8 leaving 4,5, and 9. 
And the subtraction could be 9-5=4 or 9-4=5

So here are two possibilities so far:

9-5=4      9-4=5
1+7=8  and 1+7=8     
2×3=6      2×3=6
 
The addition could be 4+5=9 leaving 1,7, and 8. 
And the subtraction could be 8-1=7.  So that's another
possibility:

8-1=7
4+5=9
2×3=6

Edwin