Question 31587
Let first integer = x 

so second integer = x + 1 

according to question we get the equation,,,,

(x) + (x + 1) < 55

x + x + 1 < 55

2x + 1 < 55

subtract 1 from both sides ,we get 

2x < 54

divide both sides by 2 we get ......

x < 27

1st integer = x  and x < 27
 also as we need sum of integer with greates sum so we take x  = 26 

and 2nd integer = x+1
               =  27 + 1 
               = 28

now if you add both the integers you get sum = 54 which is less than 55 and so this verify you second condtion also .......

hope this will help you 

Please feel free to revert back for any further queries.