Problem Solving

Hi all,

I don't understand a specific explanation to this question. Thank you for your help in advance.

Q: If 0 < x < 7 and y < 9, then the lowest possible value for xy is
A. between 60 and 70
B. between 0 and 60
c. 0
d.between 0 and -70
e. less than -70

I understand why the answer is E, but I dont understand why the explanation said: since the lowest value of x is 0, the lowest possible value of xy is also 0. I thought the lowest value of X is 1 as the sign is not equal or greater but just greater.


FYI: This is the whole explanation:" It is helpful to consider extreme values when solving inequality problems, especially when given multiple inequalities. For example, the greatest xy is the product of the highest possible value of x and the highest possible value of y. Since the greatest value of x is less than 7 and the greatest possible value of y is 9, the greatest possible product would be less than 63.
In this situation, we are searching for the lowest possible value of xy. Do not make the mistake of assuming that, since the lowest value of x is 0, the lowest possible value of xy is also 0. Because there is no lower limit to the value of y, there is no lower limit to the value of xy. For example, x could be 2 and y could be -100, and xy could therefore be -200.Alternatively, note that x is a positive number while y could be infinitely negative. Since the product of a positive number and a negative number is negative, xy could be infinitely negative.The correct answer is E.

That explanation is a little sloppy. As you indicated, x cannot be 0 as the question says that x > 0.

However, the lowest value of x is not 1 as the question does not say that x must be an integer. So x can be any number that is greater than 0, including fractional numbers between 0 and 1.

Meanwhile, even though x cannot equal 0, since x can be a fractional number infinitesimally greater than 0, in testing extremes you may want to test 0, maybe not for this question, but in handling questions that use similar boundaries in different ways.

Notice that while in this question x and y cannot actually be equal to 7 and 9 respectively, testing using those upper limits would make sense were the question asking for the greatest possible value of xy. Even though that value cannot be 63, it can be a number just less than 63.

Well, the explanation says

Do not make the mistake of assuming that, since the lowest value of x is 0, the lowest possible value of xy is also 0.

So it isn't saying that the minimum is 0, it's saying NOT to make that mistake. A lot of people assume that y < 9 means that 0 < y < 9, but it doesn't: if we only know y < 9, y could be -1, -2, -3, ..., whatever.

Thank you guys! The question is from CAT 1 in Manhattan Prep.

nsuen wrote:Thank you guys! The question is from CAT 1 in Manhattan Prep.

It's definitely a good one to discuss, I'm glad you brought it up. Assuming y < 9 means 0 < y < 9 is very common and an easy mistake to make ... which is why the GMAT likes to test it so much!