XY

[kamalakarthi]

Hi, Can you please help me on the attached.

[Brent@GMATPrepNow]

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

Let's examine the EXTREME VALUES Of x and y and see what happens.

If we want to MINIMIZE the value of xy, we need to examine what happens when 1 EXTREME value is positive and 1 EXTREME value is negative.
case a: x = -4 and y = 3, in which case xy = -12
case b: x = 7 and y = -6, in which case xy = -42
Great, so xy is MINIMIZED when x = 7 and y = -6
Of course, we're told that x < 7 and y > -6, but that's fine. Basically, this means that xy > -42

At this point, we know that the correct answer must be either A or B.

Next, if we want to MAXIMIZE the value of xy, we need to examine what happens when both EXTREME values are positive or both are negative.
case c: x = -4 and y = -6, in which case xy = 24
case d: x = 7 and y = 3, in which case xy = 21
Great, so xy is MAXIMIZED when x = -4 and y = -6
Of course, we're told that x > -4 and y > -6, but that's fine. Basically, this means that xy < 24

So, as you can see, -42 < xy < 24

Answer: B

Cheers,
Brent

[[email protected]]

Hi kamalakarthi,

I think that Brent accidentally posted an answer to a different question here.

For this question, we have two ranges of values to consider:

-4 < X < 7
-6 < Y < 3

And we're asked for the range of XY. To properly answer this question, we have to be sure that we've found the lowest and highest possibilities.

From the answer choices, we can tell that the least possible number is a negative value, but HOW negative can we get.... Multiplying two values together and ending with a negative product means that one value will be positive and one will be negative....

Looking at the "extreme" possibilities in the two ranges, that product would be (7)(-6) = -42

So -42 < XY

Now we have to go after the highest value, which will mean EITHER multiplying two positive values OR two negative values.

The possibilities are...
(7)(3) = 21
(-4)(-6) = 24

Since 24 is bigger, that value is our maximum.

The final range is....

-42 < XY < 24

Final Answer: B

GMAT assassins aren't born, they're made,
Rich

[Matt@VeritasPrep]

Given the inequalities

-4 < x < 7
-6 < y < 3

we know that the least xy = (greatest positive #) * (least negative #)
and that the greatest xy is EITHER the product of the two greatest positives OR the product of the two least negatives, whichever is greater.

Hence the least xy = -6 * 7 and the greatest xy = (-4) * (-6), so -42 < xy < 24.

[Brent@GMATPrepNow]

Thanks Rich - I wasn't paying attention.

I've fixed my post.

Cheers,
Brent

[GMATinsight]

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

CONCEPT: For maximizing the value of xy the ABSOLUTE VALUE must be as high as possible and the sign should be positive if possible
Similarly, For minimizing the value of xy the ABSOLUTE VALUE must be as high as possible and the sign should be NEGATIVE if possible

Maximum (xy) = Max[(-4)x(-6) and (7)x(3)] = 24
Minimum (xy) = Min[(-4)x(3) and (7)x(-6)] = -42

Therefore, -42<xy<24

Answer: Option B[/quote]

[GMATGuruNY]

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

To combine ranges with respect to a given operation, perform the given operation using EVERY COMBINATION OF ENDPOINTS.

Here, the given operation is MULTIPLICATION.
Calculating the value of xy using every combination of endpoints, we get:
(-4)(-6) = 24.
(-4)(3) = -12.
(7)(-6) = -42.
(7)(3) = 21.
The SMALLEST result gives us the LOWER limit of xy: -42.
The GREATEST result gives us the UPPER limit of xy: 24.
Thus:
-42 < xy < 24.

The correct answer is B.

Another example:

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

Here, the given operation is SUBTRACTION.
Calculating the value of x-y using every combination of endpoints, we get:
-4 - (-6) = 2.
-4 - 3 = -7.
7 - (-6) = 13.
7 - 3 = 4.
Thus:
-7 < x-y < 13.

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[GMATinsight]

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

The same example posted above can be looked at in this way with more certainty to get the answer logically and quickly

for (x-y) to be Maximum x must be Maximum and y must be Minimum

i.e. Max(x-y) = 7-(-6) = 13


for (x-y) to be Minimum x must be Minimum and y must be Maximum

i.e. Min(x-y) = (-4)-(3) = -7

Therefore,
-7 < (x-y) < 13