inequalities question (1 Viewer)

[sasquatch]

Is it ok to do this:

E.g. Show (a² – b²)(c² – d²) <= (ac² – bd²)

Working backwards:
(a² – b²) (c² – d²) <= (ac² – bd²)
a²c² - a²d² - b²c² + b²d² <= a²c² – 2abcd + b²d²
- a²d² - b²c² <= - 2abcd
a²d² + b²c² >= 2abcd
a²d² - 2abcd + b²c² >= 0
(ad – bc)² >= 0

Then what I write on my hsc paper:
(ad – bc)² >= 0
and then work backwards from what I wrote above.

Is it ok to do this, because the result (ad – bc)² >= 0 is actually true.. so yeah?

 

[Affinity]

Is it ok to do this:

E.g. Show (a² – b²)(c² – d²) <= (ac² – bd²)

Working backwards:
(a² – b²) (c² – d²) <= (ac² – bd²)
a²c² - a²d² - b²c² + b²d² <= a²c² – 2abcd + b²d²
- a²d² - b²c² <= - 2abcd
a²d² + b²c² >= 2abcd
a²d² - 2abcd + b²c² >= 0
(ad – bc)² >= 0

Then what I write on my hsc paper:
(ad – bc)² >= 0
and then work backwards from what I wrote above.

Is it ok to do this, because the result (ad – bc)² >= 0 is actually true.. so yeah?

Your first line doesn't follow from the second.

 

[silentprayer]

hmm yeah, u treated the RHS as a whole square

 

[acmilan]

In general, so long as all your steps are reversible its fine

 

[onebytwo]

(ad-bc)^2>=0
then expand and rearrange to get
a^2d^2+b^2c^2 >=0
then multiply throughout by -1
then add a^2c^2+b^2d^2
then factorise and rearrange

kind of like what you did but the other way around

 

[sasquatch]

Your first line doesn't follow from the second.

Woops i wrote the question wrong, sorry.

it was meant to be (ac – bd)²

(ad-bc)^2>=0
then expand and rearrange to get
a^2d^2+b^2c^2 >=0
then multiply throughout by -1
then add a^2c^2+b^2d^2
then factorise and rearrange

kind of like what you did but the other way around

I think you misunderstood me.

What i was asking was if i could do it backwards (say on another piece of paper, or in pencil on another page) and then copy that "backward working" in reverse onto the HSC paper (with a valid statement e.g. (ad-bc)^2>=0), to find the answer that i meant to show?

 

[edwardpham_2000]

yes
the best thing to do is to start with a fact

 

[Riviet]

What i was asking was if i could do it backwards (say on another piece of paper, or in pencil on another page) and then copy that "backward working" in reverse onto the HSC paper (with a valid statement e.g. (ad-bc)^2>=0), to find the answer that i meant to show?

Yes, don't forget to cross out the backwards working after copying it out forwards.

 

[sasquatch]

Yeah im planning to do it in pencil on the adjacent page