x² < 7x

x² - 7x < 0

x(x - 7) < 0

the zeros of the left side are 0 and 7,
the critical numbers.

We test a point in the three intervals:

(-∞,0), (0, 7), (7,∞)

-1(-1 - 7) < 0
         8 < 0  false, so not (-∞,0),

1(1 - 7) < 0
       -7 < 0   true, so (0,7)

8(8 - 7) < 0
       8 < 0    false, so not (7,∞)

Therefore the only integers that satisfy
the given inequality are the integers in (0,7),
which are {1,2,3,4,5,6}.

Edwin