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Is √x+x>-√y？

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Is √x+x>-√y？

by Max@Math Revolution » Mon Apr 02, 2018 11:03 pm
[GMAT math practice question]

Is √x+x>-√y？

1) √x+√y = 1
2) x>0
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Source: — Data Sufficiency |

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by Terry@ThePrincetonReview » Tue Apr 03, 2018 6:53 am
Let's examine the stem before diving into statements.

√x must always be positive because by convention the square root is always the positive square root. We also know that x must be positive; otherwise its square root would be imaginary. So the left side is always positive, because we are adding two positive numbers. On the right, √y is always positive, so -√y is always negative. Therefore, the question is asking: Yes or No, is the positive number on the left always greater than the negative number on the right? This is always true, EXCEPT when both x and y are 0. Therefore, the question can be boiled down to this "reformulated question": Yes or No, x = 0 and y = 0? (When the answer to this reformulated question is NO, then the equation is always TRUE/YES; when the answer is YES, then the question is always FALSE/NO.)

In S1, x could equal 1 OR y could equal 0; however, both can never equal 0, which is what we want to know. Therefore, the answer to reformulated question is NO and the equation is always TRUE; the statement is sufficient. We're down to A and D.

In S2, all that's known is that x is always positive, therefore never equal to 0. Nothing about the value of y is known or required. But again, the answer to the reformulated question, "Are BOTH x and y equal to 0?" is always NO and so the equation is always TRUE. Statement 2 is sufficient and the correct answer is D.
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If one did not gain this insight from studying the stem beforehand, it's still possible to get to the right answer. Combining the stem and statement 1, rewrite the equation in the stem as: √x + √y > -x, simplified to to 1 > -x. The value of -x is either 0 (when x = 0) or negative, so this reworked equation is always true. S1 is sufficient, so we're down to A and D. Next, combining the stem and statement 2, we see that the left side of the stem equation must always be positive (a positive plus a positive) and right side either 0 (when y = 0) or negative. S2 is sufficient, so the correct answer is D.
Last edited by Terry@ThePrincetonReview on Tue Apr 03, 2018 10:49 am, edited 2 times in total.
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by Terry@ThePrincetonReview » Tue Apr 03, 2018 7:08 am
What's interesting about this question is that it comes close to being "trivial" -- that is, a DS question that is always true/yes or always false/no based on the stem, without having to evaluate the statements. There is one little exception, when both x and y are 0, but the exception is critical. The GMAT, to the best of my knowledge, will never create a truly trivial DS question.

Another point: the GMAT does not truck in imaginary numbers (the square root of a negative number). However, to the best of my knowledge, the actual GMAT will set up the problem parameters so that an imaginary number can never be reached. In other words, on an actual test this question would probably be worded, "If x and y are both non-negative, then is √x + x > -√y?"

If interested, see my full solution to the question posted earlier.
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EDIT

by Max@Math Revolution » Thu Apr 05, 2018 3:16 am
=>

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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

√x+x>-√y
=> √x+√y+ x>0
Since √x+√y≥0 is always true, we just need to check if x > 0.
Condition 2) is sufficient and since condition 1) is hard to check and condition 2) is easy to check, the answer is D by CMT (Common Mistake Type) 4B.

Condition 1)

We have x ≥ 0 from √x,
Then √x+√y + x ≥ 1 > 0.
Thus condition 1) is sufficient.


Condition 2)

Since √x+√y ≥ 0 and x > 1, we have √x+√y + x > 1 > 0.
Thus condition 2) is sufficient.

Therefore, D is the answer.


Answer: D

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.
Last edited by Max@Math Revolution on Thu Apr 19, 2018 10:27 pm, edited 1 time in total.
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roots

by GMATGuruNY » Thu Apr 05, 2018 12:08 pm
Max@Math Revolution wrote:[GMAT math practice question]

Is √x+x>-√y？

1) √x+√y = 1
2) x>0
Question stem, rephrased:
Is √x + √y + x > 0?

Statement 1:
Substituting √x+√y = 1 into the rephrased question stem, we get:
1 + x > 0 ?
x > -1?
Since √x implies that x≥0, the answer to the blue question is YES.
SUFFICIENT.

Statement 2:
Substituting x = POSITIVE into the rephrased question stem, we get:
POSITIVE + √y + POSITIVE > 0 ?
√y > NEGATIVE + NEGATIVE?
√y > NEGATIVE ?
Since √y = the POSITIVE ROOT OF Y, the answer to the blue question is YES.
SUFFICIENT.

The correct answer is D.
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