EIV 4th Edition Chapter 6 Prob 10 page 101

[georgepa]

If AB/7 > 1/14 and A=B which of the following must be greater than 1

The solution states:
(C) 2A^2 since 2A^2 > 1

Fairly obvious.

Can someone please explain why A+B is not greater than 1

Thanks

[siddharth.sheth]

I will try and explain this to the best of my ability.

First of all, note that the question states which of the following MUST be greater than 1. This means, that under all possible circumstances, the values of the expressions given in the 5 choices will ALWAYS be greater than 1.

Now for choice A which states A+B: From the solution it follows that 2A^2>1
thus A> plus/minus (1/(sqrt2)). (This follows from the fact that sqrt(x^2) =|x|).
Since A = B this is also the value of B.
If you take the positive value of A, then A+B = 2A >2/sqrt(2) >sqrt(2) > approx 1.414 which is greater than 1.

But, on the other hand if you take the negative value of A which is now greater than -1/sqrt(2), then the value of A+B = 2A is now > -2/sqrt(2) which is greater than approx -1.414 which is MAY OR MAY NOT BE greater than 1.

Thus A+B MAY or MAY NOT BE > 1. For a question that states MUST, as explained above, it MUST be valid under all conditions.

Hopefully that helps.

[Ben Ku]

Siddarth's explanation is good.

We can solve the inequality this way:
AB / 7 > 1/14
AB > 1/2

If A = B, then
A^2 > 1/2
A^2 - 1/2 > 0
(A + sqrt(1/2))(A - sqrt(1/2)) > 0
[We can estimate sqrt (1/2) = (sqrt 2) / 2 = (1.4)/2 = 0.7]

(A + 0.7)(A - 0.7) > 0
So here, either A > 0.7 or A < 0.7

We don't know whether A and B are positive or negative, so if A and be are negative, then their sum would NOT be greater than 1.

If AB/7 > 1/14 and A=B which of the following must be greater than 1. Hope that helps.

[Nz]

Thanks for the nice explanation!

I think that you made a typo and A < -0.7 So it should be:

(A + 0.7)(A - 0.7) > 0
So here, either A > 0.7 or A < - 0.7

Pl correct me if I am wrong.

Thank you
Nz

[jnelson0612]

Thanks for the nice explanation!

I think that you made a typo and A < -0.7 So it should be:

(A + 0.7)(A - 0.7) > 0
So here, either A > 0.7 or A < - 0.7

Pl correct me if I am wrong.

Thank you
Nz

That looks good to me! Thanks for the catch. :-)