# 'Show that' help

#### Glitch

Suppose that ab > 0. Show that if a < b, then 1/b < 1/a.

I know this is correct, but I don't know how to show it. I did this:

aa^-2 < ba^-2

1/a < ba^-2

(b^-2)/a < (a^-2)/b

Hoping that I could find a way to swap the inequality sign around whilst creating the fractions required. But I can't seem to do it.

Any assistance would be great!

#### undefined

MHF Hall of Honor

Suppose that ab > 0. Show that if a < b, then 1/b < 1/a.

I know this is correct, but I don't know how to show it. I did this:

aa^-2 < ba^-2

1/a < ba^-2

(b^-2)/a < (a^-2)/b

Hoping that I could find a way to swap the inequality sign around whilst creating the fractions required. But I can't seem to do it.

Any assistance would be great!
You can consider two cases: (1) a and b are both positive, or (2) a and b are both negative.

#### Glitch

Ok. But am I on the right track? Is there a method to proving this?

#### Also sprach Zarathustra

You can consider two cases: (1) a and b are both positive, or (2) a and b are both negative.
We can do it without "consider two cases"...

1/b<1/a

1/b-1/a<0

{a-b}/ab<0

since ab>0, and a<b, it's obvious!

• Glitch and undefined

#### Glitch

Thanks! Looks like I was way off! If I were to write that in an exam, would I have to explain why it's obvious?

#### Also sprach Zarathustra

Yes, you explain it in the next form:

{a-b}<0 since a<b(given)
ab>0 {given}

so, {a-b}/ab={-}/{+}={-}<0

ok?

• Glitch

#### Glitch

Yup. Thanks. I really do need more practise with this stuff.

#### undefined

MHF Hall of Honor
Thanks Also sprach Zarathustra, my way was pretty tedious compared with yours.

Thanks! Looks like I was way off! If I were to write that in an exam, would I have to explain why it's obvious?
You should explain it on an exam, yes, but it's simply that the numerator is negative while the denominator is positive...

Edit: Too slow.

• Also sprach Zarathustra

#### undefined

MHF Hall of Honor
One more thing worth mentioning since "show that" means we're dealing with formal proofs here.

1/b<1/a

1/b-1/a<0

{a-b}/ab<0
These lines are connected implicitly by "if and only if" as in

1/b<1/a

$$\displaystyle \displaystyle \iff$$ 1/b-1/a<0

$$\displaystyle \displaystyle \iff$$ {a-b}/ab<0

For the last line, we don't need to think about whether it's "if and only if" because we only need to go in one direction

1/b<1/a

$$\displaystyle \displaystyle \iff$$ 1/b-1/a<0

$$\displaystyle \displaystyle \iff$$ {a-b}/ab<0

$$\displaystyle \displaystyle \Longleftarrow ab > 0 \land a < b$$

I write this mainly so that nobody gets confused, thinking the proof is not valid because we assumed what we wanted to prove.

• Also sprach Zarathustra

#### Glitch

I've been meaning to ask, what does that upside-down 'V' symbol mean?