if $m<\frac{1}{2}$, what is $-\frac{1}{m}$ equal to?

The mistake is that you are assuming $m$ positive but we need to consider two cases

$$m<\frac{1}{2} \iff \frac1m>2\iff -\frac1m<-2$$

$$m<0 \iff -\frac1m>0$$

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?

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