TITLE: if $m<\frac{1}{2}$, what is $-\frac{1}{m}$ equal to?
QUESTION [6 upvotes]: I start out with: $$m<\frac{1}{2}$$
Taking the reciprocal of both side flips inequality sign: $$\frac{1}{m}>2$$
Multiply both side by -1 flips the sign yet again: $$-\frac{1}{m}<-2$$.
But the result $-\frac{1}{m}<-2$ is not valid. If $m=-100$, then $-\frac{1}{m}=\frac{1}{100}$
What did I do wrong?

REPLY [2 votes]: Go through your steps with $m=-100$
$-100 < 2$  is that true?
So $\frac 1 {-100}> \frac 12$ is that true?
So $-\frac 1 {-100}=\frac 1 {100}<\frac 12$ is that true?
Can you see the error?
Maybe we can break this further:
$-100 < 2$ so
$\frac {-100}2 <1$ so
$\frac {1}2 <\frac 1 {-100} $ so
$\frac 1 {-100}>\frac 12$.
Can you pin point the exact point of wrror.  And why?