If -4<x<7 and -6<y<3, which

[arthi9487]

If -4<x<7 and -6<y<3, which of the following specifies all the possible values of xy?

A. -42<xy<21
B. -42<xy<24
C. -28<xy<18
D. -24<xy<21
E. -24<xy<24

I listed out the possibilities for x and y and looked at the different products possible. If my math is correct, it looks like 30<xy<15, but I can't figure out where I went wrong. The official answer is B.

[RonPurewal]

Hint: "-4 < x < 7" does NOT mean that x has to be -3, -2, -1, 0, 1, 2, 3, 4, 5, or 6.
Don't forget that whole numbers are not the only numbers out there in the world.

For instance, if you know that x > 5 and y > 4, then xy can be any number greater than 20. (Just think about what happens if x is, say, 5.00000000001 and y is 4.0000000000001.)
So, you don't even have to think about those weird decimals -- you can just plug in the boundary numbers themselves, as long as you recognize that you have to be just a little bit greater (or less) than those numbers.

[MaxF412]

Hey Ron,

I was wondering if you could explain this problem a little further. How come you pick the 7 and -6 as your lower bound and the -4 and -6 as your upper bound and left out the 3? Is it because that gives you the largest possible bounds? Thanks for your help.

[slaughterGMAT]

-4<x<7 and -6<y<3 -
Based on above conditions look at the various possibilities around the boundary -
x can be very nearly equal to -4 or 7 or in between (anything)
similarly y can be very nearly equal to -6 or 3 or in between.
take
1. x,y {-4,-6} = 24
2. x,y {-4,3} = -12
3. x,y {7,-6} = -42
4. x,y {7,3} = 21

the extreme values above are -42 and 24 - so xy can be anything in between -42 or 24 or very nearly equal to these values.
so -42<xy<24

[RonPurewal]

^^ that's an excellent explanation.

[RonPurewal]

more generally, in this type of situation, the possibilities will include both positive and negative values.

• to find the maximum value—which, in this type of situation, will be positive—you'll need to look at either
... the product of two positive boundary values, or
... the product of two negative boundary values.

• to find the minimum value—which, in this type of situation, will be negative—you'll need to look at the product of one positive boundary value and one negative boundary value.

if you think about this for a moment, you'll realize that exactly one of the four boundary values is going to be "left out" in each case. (if the maximum comes from multiplying the two positive bounds, then one of those bounds will be re-used in finding the minimum, and one of the negatives will be left out. and vice versa if the maximum comes from multiplying the two negative bounds.)

[ajaym8]

Hi Ron,

I got this question in my GMATPrep mock yesterday. Got a doubt.
When I write -4<x<7 and -6<y<3, in my understanding, it does NOT mean that x can equal -4 or 7 and y can equal -6 or 3.
I am saying this because there is a difference between -4<x<7 and -4<=x <= 7.

Are the 2 expressions in bold above same on the GMAT ?
I did not select the boundary terms during making the combinations due to above understanding and thus got confused.

Can you please help me with this ?
Also, just curious because I just wrote this above, which one of the expressions below is correct ? ::

on the GMAT
in the GMAT

I scored 790 on the GMAT
I scored 790 in the GMAT

Thanks,
ajaym8

-4<x<7 and -6<y<3 -
Based on above conditions look at the various possibilities around the boundary -
x can be very nearly equal to -4 or 7 or in between (anything)
similarly y can be very nearly equal to -6 or 3 or in between.
take
1. x,y {-4,-6} = 24
2. x,y {-4,3} = -12
3. x,y {7,-6} = -42
4. x,y {7,3} = 21

the extreme values above are -42 and 24 - so xy can be anything in between -42 or 24 or very nearly equal to these values.
so -42<xy<24

[ajaym8]

Hi Ron,

I got this question in a GMATPrep mock. Got a doubt.
Consider 2 expressions -4<x<7 and -4<= x <=7

Are the above 2 expressions same on the GMAT ?
As per my current understanding , they are different. That's why I did not choose the boundary terms (-4,7) and could not find an answer to the problem.

Also, just curious because I just typed this above, which one of the below expressions is correct ?

I scored 790 on the GMAT
I scored 790 in the GMAT

Thanks,
ajaym8

-4<x<7 and -6<y<3 -
Based on above conditions look at the various possibilities around the boundary -
x can be very nearly equal to -4 or 7 or in between (anything)
similarly y can be very nearly equal to -6 or 3 or in between.
take
1. x,y {-4,-6} = 24
2. x,y {-4,3} = -12
3. x,y {7,-6} = -42
4. x,y {7,3} = 21

the extreme values above are -42 and 24 - so xy can be anything in between -42 or 24 or very nearly equal to these values.
so -42<xy<24

[RonPurewal]

Hi Ron,

I got this question in a GMATPrep mock. Got a doubt.
Consider 2 expressions -4<x<7 and -4<= x <=7

Are the above 2 expressions same on the GMAT ?

nothing is different "on the GMAT", EVER!
this exam will ONLY use words or notation in ways that are absolutely universal.

As per my current understanding , they are different. That's why I did not choose the boundary terms (-4,7) and could not find an answer to the problem.

the point of using the boundary values is that you can get as close to those values as you want.
e.g., it's true that x can't be 7, but, x can be, for instance, 6.999999999999999999.
...so, by using those kinds of values for x, you can get as close as you want to the values you get by actually plugging in the extremes.

if this doesn't make sense, just imagine a simpler example:
say i tell you x < 5.
what can you tell me about 2x?
...you can tell me that 2x < 10. (it can be any value up to that, including values like 9.9999999999999999.)
hopefully that example isn't troublesome; this is the same thing.

[RonPurewal]

Also, just curious because I just typed this above, which one of the below expressions is correct ?

I scored 790 on the GMAT
I scored 790 in the GMAT

we say "on", but, this is completely irrelevant to the exam.