Problem Solving

gmat009 wrote: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 determine the largest possible value of xy, we either multiply together the two smallest negative values or the two largest positive values. Since (-4)(-6) = 24 and (7)(3) = 21, and 24 > 21, we see that the largest possible product of x and y is less than 24.

To determine the smallest value of xy, we multiply the largest positive number by the smallest negative number. Thus, the product of x and y must be greater than (7)(-6) = -42. Thus:

-42 < xy < 24

Answer: B

gmat009 wrote: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

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