Suppose for a moment that there was no greatest integer involved:

x/5 - x/7 = 1

7x - 5x = 35

x = 35/2

The floor function can change each term by an amount between 0 and 1.  So we should be able to put upper and lower bounds on the possible values of x.

Rewrite the equation without ⌊ ⌋ but with RHS = 0 and 2 instead of 1.

x/5-x/7=0 --> x=0  lower bound on x (or actually 1 since x must be positive)

x/5-x/7=2 --> x=35 upper bound on x.

So I expect no solutions > 35

Nevertheless I ran my short program much higher.

ans = []

for n in range(1, 1000000):

    if int(n/5) - int(n/7) == 1:

        ans.append(n)

print(ans)

Output:

[5, 6, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 28, 29]