If you take the reciprocal in an inequality, would it change the > / <...
If you take the reciprocal in an inequality, would it change the signs? Example:
In the example above, if you take the reciprocal of
would that flip the to or not? In another words, if you take the reciprocal of
would it be like this:
Answer & Explanation
Beginner2022-06-20Added 22 answers
If a and b have the same nonzero sign, then
(i.e., taking reciprocals reverses the inequality). If a and b have opposite (nonzero) signs, then
(i.e., taking reciprocals preserves the inequality). These follow from the fact that the function defined on the nonzero reals is strictly decreasing and sign-preserving on each component and of its domain. If either of a or b is zero, then you can't take reciprocals. Finally, compound inequalities like should be separated into " and " and each component considered separately. I also remark that inverting a sum is not the same as inverting the addends separately.
Beginner2022-06-21Added 4 answers
It depends if and are the same sign. Case 1: then and and so If both positive, flip. Case 2: then . Then as we flip when we do so so . Don't flip. Case 3: then and . Flip if they are the same sign. But FOR THE LOVE OF GOD!!!!!!! the reciprical of is !!!!!!!NOT!!!!!! !!!!!!!! It is
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