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Inequalities- data sufficiency

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Source: — Data Sufficiency |
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by GMATGuruNY » Fri Nov 21, 2014 2:41 am
The question stem should read as follows:
If y≠0, is x³/y = x + x/y?

a) y = 8
b) y = x² - 1
Statement 1: y=8
Substituting y=8 into x³/y = x + x/y, we get:
x³/8 = x + x/8
x³ = 8x + x
x³ = 9x.

Question stem, rephrased:
Is x³ = 9x?
If x=0, the answer is YES.
If x=1, the answer is NO.
INSUFFICIENT.

Statement 2: y = x² - 1
Case 1: x=2, y=3
Substituting x=2 and y=3 into x³/y = x + x/y, we get:
2³/3 = 2 + 2/3
2³ = 6 + 2
8=8.
In this case, x³/y = x + x/y, so the answer to the question stem is YES.

Test an EXTREME case.

Case 2: x=10, y=99
Substituting x=10 and y=99 into x³/y = x + x/y, we get:
10³/99 = 10 + 10/99
10³ = 990 + 10
1000=1000.
In this case, x³/y = x + x/y, so the answer to the question stem is YES.

Since the answer is YES in both cases, SUFFICIENT.

The correct answer is B.
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by GMATGuruNY » Fri Nov 21, 2014 6:11 am
If y≠0, is x³/y = x + x/y?

a) y = 8
b) y = x² - 1
Algebraic approach:

Multiplying each side of the question stem by y, we get:
x³ = xy + x
x³ - xy - x = 0
x(x² - y - 1) = 0.
The resulting equation is valid if x=0 or if x² - y - 1 = 0.

Question stem, rephrased:
Does x=0 or x² - y - 1 = 0?

Statement 1: y=8
If x=0, then the answer to the rephrased question stem is YES.
If x=1, then the answer to the rephrased question stem is NO.
INSUFFICIENT.

Statement 2: y = x² - 1
Thus:
0 = x² - y - 1.
Since x² - y - 1 = 0, the answer to the rephrased question stem is YES.
SUFFICIENT.

The correct answer is B.
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by gmat_thingie » Fri Nov 21, 2014 6:31 am
Hi,

Thanks for your detailed reply.The question stem is clearly devoid of any fact that y is not equal to zero.

If so, does the answer change? I am checking that answer is mentioned to be 'C' then. Kindly explain that.
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by GMATGuruNY » Fri Nov 21, 2014 6:59 am
gmat_thingie wrote:Hi,

Thanks for your detailed reply.The question stem is clearly devoid of any fact that y is not equal to zero.

If so, does the answer change? I am checking that answer is mentioned to be 'C' then. Kindly explain that.
if it's possible that y=0, we get:

Statement 2: y = x² - 1
Case 1: x=1 and y=0
In this case, the equation in the question stem is undefined, so the answer to the question stem is NO.
Case 2: x=2 and y=3
In this case, x³/y = x + x/y, so the answer to the question stem is YES.
Since the answer is NO in Case 1 but YES in Case 2, INSUFFICIENT.

Statements combined:
Only two cases are possible:
x=3 and y=8.
x=-3 and y=8.
In each case, x³/y = x + x/y, so the answer to the question stem is YES.
SUFFICIENT.

The correct answer is C.

Generally, the GMAT precludes division by 0.
If this problem were to appear on the GMAT, the prompt would almost certainly make clear that y≠0.
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by gmat_thingie » Fri Nov 21, 2014 7:02 am
Perfect. Thanks for your reply
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by Mathsbuddy » Fri Nov 21, 2014 7:44 am
Rearranged,x³/y = x + x/y becomes:
y = x² - 1

Statement a) If y = 8, then 8 = x² - 1, so x = 3 or -3.
We have no info about x, so we don't know if this complies. NOT SUFFICIENT.

Statement b) y = x² - 1 matches our previous rearranged equation. SUFFICIENT.
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by GMATGuruNY » Fri Nov 21, 2014 7:54 am
Mathsbuddy wrote:Rearranged,x³/y = x + x/y becomes:
y = x² - 1
Careful!
When you simplified, you divided by x.
If x=0, then this division is not allowed.
Moreover, if x=0, then x³/y = x + x/y is true for ANY NONZERO VALUE OF Y.
The following rephrasing of the question stem would be correct:
Does y = x² - 1 OR does x=0?
Last edited by GMATGuruNY on Fri Nov 21, 2014 7:56 am, edited 1 time in total.
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by Mathsbuddy » Fri Nov 21, 2014 7:55 am
gmat_thingie wrote:Hi,

Thanks for your detailed reply.The question stem is clearly devoid of any fact that y is not equal to zero.

If so, does the answer change? I am checking that answer is mentioned to be 'C' then. Kindly explain that.
The axiom that y is not zero, is purely there to eliminate dividing by zero.
In my post above, we have shown that y = x² - 1 complies for the situation when everything has been multiplied by y. At this point (only AFTER multiplying by y) x can be any value, including zero, as the equation is a generic truth for all values of x and possible values of y. However, if we include y = 0 BEFORE multiplying by y, we could get INFINITY = INFINITY, but strictly speaking this is not acceptable. Dividing by 0 is a big no-no. Therefore the question must state that y is not zero.
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by Mathsbuddy » Fri Nov 21, 2014 8:11 am
GMATGuruNY wrote:
Mathsbuddy wrote:Rearranged,x³/y = x + x/y becomes:
y = x² - 1
Careful!
When you simplified, you divided by x.
If x=0, then this division is not allowed.
Moreover, if x=0, then x³/y = x + x/y is true for ANY NONZERO VALUE OF Y.
The following rephrasing of the question stem would be correct:
Does y = x² - 1 OR does x=0?
Thanks Mitch,

100% true. I hadn't overlooked it, but I did dismiss x = 0 as trivial in this particular data sufficiency question. I'll try again below:

Rearranged,x³/y = x + x/y becomes:
xy = x(x² - 1 )

Statement a) If y = 8, then 8x = x(x² - 1), so x = 0, 3 or -3.
We have no info about x, so we don't know if this complies. NOT SUFFICIENT.

Statement b) y = x² - 1 complies with xy = x(x² - 1 ), for both x = 0 and x<>0. SUFFICIENT

So, clearly, x = 0 is indeed trivial here as we are deciding data sufficiency, rather than solving an algebra problem. However, I admit I should have expressed this in the solution, instead of hold it in my head.

Thanks for picking up on this. It is an important issue.
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