6

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?

user
  • 154,566
most venerable sir
  • 1,689
  • 1
    A hint to finding the mistake on your own: You already started off well - finding a counterexample. Next you can test your example against each of your steps to figure out where and why you went wrong. In your case, already the second step would fail and you can probably figure out why it did. – Qi Zhu Aug 30 '18 at 20:45
  • 1
    Why does taking a reciprical flip the inequality sign? Does it always? – fleablood Aug 30 '18 at 20:48

2 Answers2

8

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

  • for $0<m<\frac12$

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

  • for $m<0$

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

user
  • 154,566
2

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?

fleablood
  • 124,253