we can solve
7/9 > x/13
for x.
multiply both sides by 13:
7/9 > x/13
7/9*13 > x/13 * 3
7/9*13 > x
91/9 > x
reverse it to get
x < 91/9
this says that x has to be less than 91/9 for the inequality 7/9 > x/13 to be true.
91/9 in decimal form is 10.1 repeating.
the greatest integer x is therefore 10
answer is 10
can verify:
7/9 > x/13
7/9 is approximately 0.777777 so
0.7777 > x / 13
for x = 10, we have 10/13 = 0.7692. and 0.7777 is bigger than that
try next integer. at x = 11, we have 11/13 = 0.8462. this is bigger than 0.7777
so the answer is 10. that's the greatest integer