Question

find three consecutive even intergers for which the sum of the first two integers is greater than 9 more than the third integer, and the sum of the first and third integers is less than 17 more than the second integer

Answer 1

"find three consecutive even intergers"
Let M be the middle integer.
The three numbers are M-2, M, M+2

"the sum of the first two integers is greater than 9 more than the third integer"
(M-2 + M) > M+2 + 9
on solving
M > 13

"sum of the first and third integers is less than 17 more than the second integer"
M-2 + M+2 < M+17
on solving
M < 17

That leaves us with two choices,
M=14, or M=16.

Check:
M=14
12+14=26 > 16+9=25 OK
12+16=28 < 14+17=31 OK

M=16
14+16=30 > 18+9=27 OK
14+18=32 < 16+17=33 OK

So there are two valid solutions,
12,14,16 or 14,16,18