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Question
Find all pairs of consecutive odd positive integers, both of which are smaller than 10, such that their sum is more than 11.
Solution
Let the smaller odd positive integer be x. Then, the other odd positive integer shall be x + 2.
Therefore, as per the given conditions:
\[x + 2 < 10 \text{ and } x + x + 2 > 11\]
\[ \Rightarrow x < 8 \text{ and } 2x > 9\]
\[ \Rightarrow x < 8 \text{ and } x > \frac{9}{2}\]
\[\text{ Since x is an odd integer }, \]
\[\text{ Therefore }, x = 5, 7\]
\[\text{ Hence, pairs are } (5, 7), (7, 9) .\]
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