Inequality

[navdeep_bajwa]

MGMAT Challenge Problems on 04/11/05

w, x, y, and z are integers. If ,z>y>x>w is abs w > x^2 > abs y > z^2 ?

1) wx>xz

2) zx>wy
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

In complex and abstract Data Sufficiency questions such as this one, the best approach is to break the question down into its component parts.

First, we are told that z > y > x > w, where all the unknowns are integers. Then we are asked whether it is true that . Several conditions must be met in order for this inequality to be true in its entirety:

(1) abs y > z^2
(2) x^2 > abs
(3) y > z^2


In order to answer "definitely yes" to the question, we need to establish that all three of these conditions are true. This is a tall order. But in order to answer "definitely no", we need only establish that ONE of these conditions does NOT hold, since all must be true in order for the entire inequality to hold. This is significantly less work. So the better approach in this case is to see whether the statements allow us to disprove any one of the conditions so that we can answer "definitely no".

But in what circumstances would the conditions not be true?

Let's focus first on condition (1): . Since z > y, the only way for to be true is if y is negative. If y is positive, z must also be positive (since it is greater than y). And taking the absolute value of positive y does not change the size of y, but squaring z will yield a larger value. So if y is positive, must be larger than the absolute value of y.

If you try some combinations of actual values where both y and z are positive and z > y, you will see that is always true and that is never true. For example, if z = 3 and y = 2, then is true because . But if z = 3 and y = -10, then is true because . The validity of depends on the specific values (for example, it would not hold true if z = 3 and y = -1), but the only way for to be true is if y is negative.

And if y must be negative, then x and w must be negative as well, since y > x > w. So if we could establish that any ONE of y, x, or w is positive, we would know that is NOT true and that the answer to the question must be "no".

Statement (1) tells us that wx > yz. Does this statement allow us to determine whether y is positive or negative? No. Why not? Consider the following:

If z = 1, y = 2, x = -3, and w = -4, then it is true that wx > yz, since (-4)(-3) > (2)(1).

But if z = 1, y = -2, x = -3, and w = -4, then it is also true that wx > yz, since (-4)(-3) > (-2)(1).

In the first case, y is positive and the statement holds true. In the second case, y is negative and the statement still holds true. This is not sufficient to tell us whether y is positive or negative.

Statement (2) tells us that zx > wy. Does this statement allow us to determine whether y is positive or negative? Yes. Why? Consider the following:

If z = 4, y = 3, x = 2, and w = 1, then it is true that zx > wy, since (4)(2) > (1)(3).

If z = 3, y = 2, x = 1, and w = -1, then it is true that zx > wy, since (3)(1) > (-1)(2).

If z = 2, y = 1, x = -1, and w = -3, then it is true that zx > wy, since (2)(-1) > (-3)(1).

In all of the cases above, y is positive. But if we try to make y a negative number, zx > wy cannot hold. If y is negative, then x and w must also be negative, but z can be either negative or positive, since z > y > x > w. If y is negative and z is positive, zx > wy cannot hold because zx will be negative (pos times neg) while wy will be positive (neg times neg). If z is negative, then all the unknowns must be negative. But if they are all negative, it is not possible that zx > wy. Since z > y and x > w, the product zx would be less than wy. Consider the following:

If z = -1, y = -2, x = -3, and w = -4, then zx > wy is NOT true, since (-1)(-2) is NOT greater than (-4)(-3).

Since y is positive in every case where zx > wy is true, y must be positive. If y is positive, then cannot be true. If cannot be true, then cannot be true and we can answer "definitely no" to the question.

Statement (2) is sufficient.

The correct answer is B: Statement (2) alone is sufficient but statement (1) alone is not.


Can you please explain how second part is true
zx > wy

if y=2,z=20,w=-8,x=-2