True or False?
For all real numbers with and being non-zero, if , then must be true.
Why some people say it's true:
It looks intuitive: for this problem, cross multiply is the same as multiplying both sides by . Then which implies .
Why some people say it's false:
There are other cases to consider besides positive real numbers. In those other cases, this "identity" might fail with one of being negative.
This statement is . The claim is true if and only if the denominators are both positive or negative. In particular, for and , while the constraint of is indeed fulfilled, the claim of is false because is not true.
The reason our initial claim fails is because once we multiply both sides of an inequality by a negative number, the inequality sign must be flipped. As an explicit example, the inequality is obviously true. But if we multiply both sides by , while keeping the inequality sign the same, we have which is obviously false.
Rebuttal: Because and makes both and true.
Reply: We have only shown that it's true when and . In fact, we should prove (or disprove) that it's true in general. That is, we did not prove that the claim holds for all reals and with .
Rebuttal: If we square both sides of the inequality, we get . Then we can cross multiply both sides by , which is a positive number, so the inequality sign does not need to be changed. Thus . Taking the square roots gives the desired claim, .
Reply: Recall that . So taking the square roots only yields , instead of .
Want to make sure you've got this concept down? Try these problems: