Question 1163645
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The possible pairs are (x,45-x)  with  the conditions

    x is whole, 45 -x is whole and  | x - (45-x) | < 10.      (1)


I use the absolute value inequality, based on the context, saying "their difference", which means both possible differences.



In other words, the numbers are (x,45-x), where

    x is integer, 0 <= x <= 45  and  -10 < 2x -45 < 10.       (2)



The last compound inequality means

     -10 + 45 < 2x < 10 + 45.

     35       < 2x < 55

     17.5     < x < 27.5.                                     (3)


There are 10 integer solutions for the last compound inequality

     x = 18, 19, 20, 21, 22, 23, 24, 25, 26, 27.



They create 10 pairs

     (18,27), (19,26), (20,25), (21,24), (22,23), (23,22), (24,21), (25,20), (26,19), (27,18).



<U>ANSWER</U>.  There are 10 pairs satisfying given conditions, listed above.
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Solved.