Inequalities
If are non-zero real numbers such that and , then which of the following is correct?
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1 solution
Moderator note:
The first half where you prove by contradiction that is negative is great.
In the second half, you have not proven that can be either positive or negative. Thus far, all that you have is "I do not have any evidence to prove that b cannot be positive or that b cannot be negative". To show that " can be either positive or negative", you will have to demonstrate it using numerical examples explicitly.
Let's assume that a is a negative number. Since |a|>|b| is true, be should be either a positive or negative number whose absolute value is less than a. In both cases, b>a is true, which is contradicting the condition of the problem. Therefore, a is a positive number. If a is a positive number satisfying |a|>|b|, b can have any sign, plus or minus. For example, let's assume that a is a positive real number n greater than 0. Then, if b is ±n/2, both condition holds. Therefore, a is positive, and b is either positive or negative.