M02-19

Official Solution:


Is \(x \gt y^2\)?

(1) \(x \gt y+5\).

Rewrite as \(x-y \gt 5\). This is clearly insufficient, for example: if \(x=1\) and \(y=-10\), then the answer is NO, but if \(x=10\) and \(y=1\), then the answer is YES. We have two different answers, so this is not sufficient.

(2) \(x^2-y^2=0\).

Rewrite as \((x-y)(x+y)=0\): so either \(x-y=0\) or \(x+y=0\). This is also insufficient: if \(x=1\) and \(y=1\), then the answer is NO, but if \(x=\frac{1}{2}\) and \(y=\frac{1}{2}\), then the answer is YES. We have two different answers, so this is not sufficient.

(1)+(2) Since we know from (1) that \(x-y \gt 5\), we can conclude that \(x-y \neq 0\). Therefore, based on (2), we must have \(x+y=0\). This means that \(y=-x\), and we can rewrite the question as: is \(x \gt x^2\)?

Now, let's substitute \(y=-x\) in (1): \(x \gt -x + 5\). From this inequality, we can infer that \(x \gt \frac{5}{2}\). Since \(x \gt \frac{5}{2}\), it implies that \(x^2 \gt x\) (because for any \(x\) greater than 1, \(x^2 \gt x\)). Therefore, the answer to the question of whether \(x \gt y^2\) is NO. Sufficient.


Answer: C
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Is \(x > y^2\)?

1) x > y+5
Instead of working with the inequality, assume that x = y+5 and keep in mind that x can have a slightly greater magnitude than that estimated (for instance +0,1) or a definitely greater one (for instance +5). Pick some values for y and compute x. If y = 1; x = 6 (precisely anything > 6) then x is greater than \(y^2\). Pick y = -6; x = -1 then \(y^2\) is greater than x. Not sufficient.

2) \(x^2 - y^2 = 0\) → x = ± y. Pick some values for x. If x = 5; y = ± 5 then x is smaller than \(y^2\). If x = 0,5; y = ± 0,5 then x is greater than \(y^2\). Not sufficient.

1+2) plug x=-y into the first inequality, you will get that y < -2,5. Say that y = -2,6 then x =2,6 and x is always smaller than \(y^2\). If two numbers are equal the question translates into: is \(x > x^2\)? Which happens only when 0 < x < 1.

Sufficient → answer C.
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I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.

I see you Bunnel. Solid Q. Respect.

How Bunuel gets numbers to plug easily. If I start plugging numbers it takes me more than 5 mins. How can I get good at this?

Number plugging:

How to Do Math on the GMAT Without Actually Doing Math
The Power of Estimation for GMAT Quant
How to Plug in Numbers on GMAT Math Questions
Number Sense for the GMAT
Can You Use a Calculator on the GMAT?
Why Approximate?
GMAT Math Strategies — Estimation, Rounding and other Shortcuts
The 4 Math Strategies Everyone Must Master, Part 1 (1. Test Cases and 2. Choose Smart Numbers.)
The 4 Math Strategies Everyone Must Master, part 2 (3. Work Backwards and 4. Estimate)
Intelligent Guessing on GMAT
How to Avoid Tedious Calculations on the Quantitative Section of the GMAT
GMAT Tip of the Week: No Calculator? No Problem.
The Importance of Sorting Answer Choices on the GMAT

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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I think this is a high-quality question. Great!

I think this is a high-quality question and I agree with explanation. Brilliant stuff Bunuel

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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I think this is a high-quality question and I agree with explanation.
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