The given inequality is xy < 5x="" +="" />

We can rewrite this inequality as xy - 5x - 6y < />

Now, let's try to factorize the left-hand side of the inequality.

xy - 5x - 6y = (x - 6)(y - 5) - 30.

So, the inequality becomes (x - 6)(y - 5) - 30 < />

Now, we can proceed to solve this inequality.

For (x - 6)(y - 5) - 30 < 0="" to="" be="" true,="" either="" (x="" -="" 6)="" and="" (y="" -="" 5)="" must="" have="" the="" same="" sign="" and="" (x="" -="" 6)(y="" -="" 5)="" ≠="" 0,="" or="" (x="" -="" 6)="" and="" (y="" -="" 5)="" must="" have="" opposite="" />

Case 1: (x - 6) and (y - 5) have the same sign and (x - 6)(y - 5) ≠ 0.
In this case, (x - 6)(y - 5) - 30 < 0="" becomes="" (x="" -="" 6)(y="" -="" 5)="" />< />

We can consider the factors of 30: ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30.

If (x - 6)(y - 5) < 30,="" we="" can="" test="" each="" />

(i) If (x - 6)(y - 5) = ±1, there are no integer solutions for x and y.
(ii) If (x - 6)(y - 5) = ±2, there are no integer solutions for x and y.
(iii) If (x - 6)(y - 5) = ±3, there are no integer solutions for x and y.
(iv) If (x - 6)(y - 5) = ±5, there are two possible solutions: (x - 6) = ±5, (y - 5) = ±1, which gives x = 1, 11 and y = 4, 6.
(v) If (x - 6)(y - 5) = ±6, there are no integer solutions for x and y.
(vi) If (x - 6)(y - 5) = ±10, there are no integer solutions for x and y.
(vii) If (x - 6)(y - 5) = ±15, there are no integer solutions for x and y.
(viii) If (x - 6)(y - 5) = ±30, there are no integer solutions for x and y.

So, in this case, there are four possible solutions: (1, 4), (1, 6), (11, 4), and (11, 6).

Case 2: (x - 6) and (y - 5) have opposite signs.
In this case, (x - 6)(y - 5) - 30 < 0="" becomes="" (x="" -="" 6)(y="" -="" 5)="" /> 30.

Since (x - 6)(y - 5) > 30, this means both factors must be positive or both factors must be negative.

If