Let's define two consecutive integers as $x$ and $x+1$.

The sum of these two integers is: $x + (x + 1) = 2x + 1$ We are told that their sum is less than 55: $2x + 1 < 55$ Now, solve for $x$: $2x < 54 \quad \text{(Subtracting 1 from both sides)}$ $x < 27 \quad \text{(Dividing both sides by 2)}$ Thus, the greatest integer $x$ must be less than 27, so the largest possible integer is $x = 26$.

Therefore, the two consecutive integers are: $26 \quad \text{and} \quad 27$

The sum of 26 and 27 is: $26 + 27 = 53$

Thus, the pair of integers with the greatest sum is $26$ and $27$, and their sum is $53$, which is less than 55.

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Would you like more details on any part of the solution? Here are some related questions you might find useful:

1. What happens if the sum is exactly 55?
2. How would the problem change if we considered the sum of three consecutive integers?
3. Can we apply this approach to any general number instead of 55?
4. What if the consecutive numbers had to be negative?
5. How does the approach change if the condition were "greater than" instead of "less than"?

Tip: When dealing with consecutive integers, expressing them algebraically as $x$ and $x+1$ (or more for larger sequences) simplifies the problem-solving process.