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smc

prealgebra senior

Problem

Real numbers , , and satisfy the inequalities , , and . Which of the following numbers is necessarily positive?
(A)
(B)
(C)
(D)
(E)
Solution
Notice that must be positive because . Therefore the answer is . The other choices: As grows closer to , decreases and thus becomes less than . can be as small as possible (), so grows close to as approaches . For all , , and thus it is always negative. The same logic as above, but when this time.
Final answer
E