What is the greatest integer x for which 7/9 > x/13? (7/9 and x/13 are both fractions)

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

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