1970 AHSME Problems/Problem 17
Problem
If , then for all and such that and , we have
Solution
If and , we can divide by the positive number and not change the inequality direction to get . Multiplying by (and flipping the inequality sign because we're multiplying by a negative number) leads to , which directly contradicts . Thus, is always false.
If (which is possible but not guaranteed), we can divide the true statement by to get . This contradicts . Thus, is sometimes false, which is bad enough to be eliminated.
If , then the condition that is satisfied. However, and , so is false for at least this case, eliminating .
If , then is also satisfied. However, , so is false, eliminating .
All four options do not follow from the premises, leading to as the correct answer.
See also
1970 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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