Problem Solving

If -2<a<11 and 3<b<12, then which of the folloing is NOT true?

A). 1<a+b<23
B). -14<a-b<8
C). -7<b-a<14
D). 1<b+a<23
E). -24<ab<132

Answer is C. and for me the only question here is what to do with E. to be true, because for me it is not true the same like C.

Why do you think it is E ?

If -2<a<11 and 3<b<12, then -24<ab<132,
let us say a=-1.9 and b =11.9 , ab = -22.61 !
let a=10.9 and b =11.9 then ab =129.71 !
So, -24<ab<132 is correct !

C). -7<b-a<14 - Let a = 10.9 and b =3.1, b-a =3.1-10.9=-7.8 -7.8 is < -7 So, this inequation is NOT true!

Yeah, keep in mind that to make ab as small as possible, it doesn't necessarily mean you should make a and b both as small as possible. That would make sense if both a and b had to be positive, but since we have negative values to consider, the smallest value of ab comes when ab is most negative. In this case it means making b as big as possible, and making a as negative as possible.

lenagmat wrote: [spoiler] for me the only question here is what to do with E. to be true, because for me it is not true

-2 < a < 11 ------------- (1)
3<b<12 ------------------ (2)
so for a consider 2 cases
(A)
-2< a < 0 in that case reverse the sign to make middle term positive
=> 0 < -a < 2
remember .. -a is positive here
now you can multiply it with second inequality to check the boundaries
=>
0< -ab < 24
now reverse the sign => -24 < ab < 0 --- (C)
(B)
0<a<11 here a is definitely positive ...
you can multiply it directly with 2
=>
0<ab <132 - ----(D)
from C and D and we also didn't include case a = 0 so ab can be 0 too if a is zero

-24 < ab <132
so E is true

As long as we can pick two allowed values and make a statement TRUE, we know it cannot be correct.

For E, the biggest a must be smaller than 11 and the biggest b must be smaller than 12. 11 x 12 = 132, so the product MUST be smaller, true.

The smallest a is greater than -2, and the smallest b is greater than 3, so the smallest ab must be greater than -6. All values greater than -6 are certainly greater than -24.