If 3 < a < 7 < b, which of the following must be greater than 20?
The product of a and b (ab) must be greater than 20.
Given the inequality 3 < a < 7 < b, a and b are both greater than 3 and 7 respectively. Multiplying two values that are greater than 3 and 7 will result in a value that is greater than 20.
The square of a (a²) will not necessarily be greater than 20. Given the inequality, a is greater than 3 but less than 7. The maximum value of a² would be 49 if a was equal to 7, but a is less than 7. Hence, a² will always be less than 49 and not necessarily greater than 20.
The value 2b will not necessarily be greater than 20. Given the inequality, b is greater than 7. The minimum value of 2b would be 14 if b was equal to 7, but b is greater than 7. Hence, 2b will always be greater than 14 but not necessarily greater than 20.
The product of a and b (ab) will be greater than 20. Given that a is greater than 3 and b is greater than 7, the minimum value of ab would be 21 if a was equal to 3 and b was equal to 7. However, both a and b are greater than 3 and 7 respectively, making the product ab always greater than 20.
The sum of a and b (b + a) will not necessarily be greater than 20. Given the inequality, a and b are greater than 3 and 7 respectively. The minimum value of b + a would be 10 if a was equal to 3 and b was equal to 7. However, both a and b are greater than 3 and 7 respectively, making the sum b + a always greater than 10 but not necessarily greater than 20.
Given the inequality 3 < a < 7 < b, the only option that must always be greater than 20 is the product of a and b (ab). While the other options can also yield values greater than 20, they do not necessarily always result in values above this threshold as per the given constraints.
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