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Is x>0?

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Is x>0?

by Max@Math Revolution » Tue Apr 10, 2018 12:30 am

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[GMAT math practice question]

$$Is\ x>0?$$

$$1)\ (x+y)^2>(x-y)^2$$
$$2)\ x+y>x-y$$
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Source: — Data Sufficiency |
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by GMATGuruNY » Tue Apr 10, 2018 4:02 am

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Max@Math Revolution wrote:[GMAT math practice question]

$$Is\ x>0?$$

$$1)\ (x+y)^2>(x-y)^2$$
$$2)\ x+y>x-y$$
Statement 1: (x+y)² > (x-y)²
x² + y² + 2xy > x² + y² - 2xy
2xy > -2xy
4xy > 0
xy > 0.
Implication:
x and y have the SAME SIGN.
Since x and y could both be positive or both be negative, INSUFFICIENT.

Statement 2: x+y > x-y
y > -y
2y > 0
y > 0.
No information about x.
INSUFFICIENT.

Statements combined:
Since y>0, and x and y have the same sign, x>0.
Thus, the answer to the question stem is YES.
SUFFICIENT.

The correct answer is C.
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by Max@Math Revolution » Thu Apr 12, 2018 12:40 am

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Condition 1) tells us that
(x+y)^2 > (x-y)^2
=> x^2 + 2xy + y^2 > x^2 - 2xy + y^2
=> 2xy > -2xy
=> 4xy > 0
=> xy > 0

Condition 2) tells us that
x + y > x - y
=> y > -y
=> 2y > 0
=> y > 0.

Since xy > 0 and y > 0, we have x > 0.

Thus, both conditions 1) & 2) together are sufficient.


In general, there are many questions involving integers and statistics to which we need to apply CMT(Common Mistake Type) 4.


Condition 1):
If x = -2 and y = -1, then the answer is "yes".
If x = 1 and y = 2, then the answer is "no".

Thus, condition 1) is not sufficient.


Condition 2):
If x = -3 and y = -1, then the answer is "yes".
If x = -1 and y = -3, then the answer is "no".
Thus condition 2) is not sufficient.

Therefore, C is the answer.

Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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by Brent@GMATPrepNow » Thu Apr 12, 2018 8:38 am

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Max@Math Revolution wrote:[GMAT math practice question]

$$Is\ x>0?$$

$$1)\ (x+y)^2>(x-y)^2$$
$$2)\ x+y>x-y$$
Target question: Is x POSITIVE?

Statement 1: (x + y )² > (x - y)²
Take: (x + y )² > (x - y)²
Expand and simplify both sides to get: x² + 2xy + y ² > x² - 2xy + y ²
Subtract x² from both sides: 2xy + y ² > -2xy + y ²
Subtract y² from both sides: 2xy > -2xy
Add 2xy both sides: 4xy > 0
Divide both sides by 4 to get: xy > 0
If the product xy> 0, then there are two possibilities:
Case a: x is positive and y is positive. In this case, the answer to the target question is YES, x is positive
Case b: x is negative and y is negative. In this case, the answer to the target question is NO, x is not positive
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x + y > x - y
Take: x + y > x - y
Add y to both sides: x + 2y > x
Subtract x from both sides: 2y > 0
Divide both sides by 2 to get: y > 0
So, we know that y is positive, but we have no information about x
So, we cannot answer the target question with certainty.
Statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that there are two possible cases (case a: x and y are both positive OR case b: x and y are both negative)
Statement 2 tells us that y is positive
So, statement 2 ELIMINATES case b, which means x and y are both positive
The answer to the target question is YES, x is positive
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

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by Scott@TargetTestPrep » Mon Apr 16, 2018 3:44 pm

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Max@Math Revolution wrote:[GMAT math practice question]

$$Is\ x>0?$$

$$1)\ (x+y)^2>(x-y)^2$$
$$2)\ x+y>x-y$$
Statement One Alone:

(x + y)^2 > (x - y)^2

x^2 + 2xy + y^2 > x^2 + y^2 - 2xy

4xy > 0

xy > 0

We see that x and y are either both positive or both negative.

Statement one alone is not sufficient to answer the question.

Statement Two Alone:

x + y > x - y

Simplifying, we have:

2y > 0

y > 0

We see that y > 0; however, since we don't know anything about x, statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using our two statements we see that xy > 0 and that y > 0; thus, x must also be greater than zero.

Answer: C

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