If |x|×y+ 9 > 0, and x and y are integers, is x < 6?

[mjfsutherland]

Sorry if this has already been answered on the forum but I couldn't find a way of searching for it as the search engine throws out all the x y terms.

If |x|×y+ 9 > 0, and x and y are integers, is x < 6?

(1) y is negative

(2) |y| < 1

OA=E

The explanation on the ManhattenGMAT CAT is:

(1) INSUFFICIENT: We know that |x|×y > -9 and that y is a negative integer. Suppose y = -1. Then |x|×(-1) > -9, which means |x| < 9 (since dividing by a negative number reverses the direction of the inequality). Thus x could be less than 6 (for example, x could equal 2), but does not have to be less than 6 (for example, x could equal 7).

My Question is how do they get rid of the x term.

If

|x|xy > -9

and we let y=-1

(-1)|x|x > -9 (divide both sides by -1)

(-1)/(-1)|x|x > 9/(-1) gives |x|x < 9 but in the answer they say |x| < 9

If |x|x < 9 then x < 3 when y=-1. As (1) tells us that y is negative and an integer we can substitute other values of y.

y=-2, |x|x < 9/2 (anything less than y=-1, x must be negative)
y=-3, |x|x < 3
y=-9, |x|x < 1
y=-20, |x|x < 9/20

which is SUFFICIENT as all we need to find is if x < 6

[mjfsutherland]

ok ok I see NOW its not an X!

|x| * y + 9

[Ben Ku]

please post if there are other questions about this problem. Thanks.

[asharma8080]

Does this algebraic approach make sense on this problem?

Given
|x| (y) > -9
Or
-(x)*(y) > -9
==> xy < 9
Also,
xy > -9

Thus we have -9 < xy < 9

Now, with statement 1, y < 0
that means:
- 9 < x (- some int) < 9
or each separately,
- 9 < - x
(some int)*9 > x
AND
- x < 9
or x > -9 (some int)

Thus, x can be < 6 or > 6. NS.

Similarly, statement 2, n.s.

E.

[RonPurewal]

Does this algebraic approach make sense on this problem?

Given
|x| (y) > -9
Or
-(x)*(y) > -9
==> xy < 9
Also,
xy > -9

Thus we have -9 < xy < 9

no, this is incorrect, as you can see by just thinking about big numbers.
for instance, if x = y = 1,000,000, then it's pretty clear that |x|y = 1 trillion is greater than -9.
in this case -9 < xy < 9 is definitely not true, so your work doesn't work here.

the problem is that you've combined two statements that aren't both true.
i.e., you've tried to let |x| equal both x and -x at the same time. that's not a thing.

--

[RonPurewal]

perhaps the most important thing you can keep in mind here is the purpose of gmat inequality problems.

MOST gmat inequality problems are specifically engineered so that plain algebra will not solve them.

inequalities are one of those topics that's specifically on here to mess with the heads of "algebra jocks" -- you'll almost always have to test cases/signs/etc.
(coordinate geometry is another such topic. very few coordinate geometry problems rely on the whole "y = mx + b" concept; most of them, especially in data sufficiency, depend heavily on visualizing the situation(s) described in the problem.)