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If x is an integer, what is the value of x^2 + x ?

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If x is an integer, what is the value of x^2 + x ?

by gmattesttaker2 » Thu Feb 13, 2014 9:45 pm
Hello,

Can you please tell me if my approach is correct here:

If x is an integer, what is the value of x^2 + x ?

(1) x^2 = |x|

(2) x^2 + x > 0

OA: C


I tried to solve as follows:

1) x^2 = |x|

Hence, x^2 + x = |x| + x

If x > 0 => |x| + x = 2x

If x < 0 => |x| + x = 0

Hence, In-suff.


2) x^2 + x > 0

In-suff.


1 and 2:

x^2 + x = |x| + x > 0
=> x cannot be negative
Hence, |x| + x = 2x

I was wondering if this solution is correct?

Thanks a lot,
Sri
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by Patrick_GMATFix » Fri Feb 14, 2014 1:04 am
Sri,

Be careful: When a DS question asks you for the value of an expression, SUFFICIENT information means that you have enough to know the exact numerical value. So in general, proving that |x|+x = 2x is not enough to say that you have sufficient info to find |x|+x, unless you know exactly what number it is equal to.

For instance, if a DS asked "what is the value of |x|+x?" and a statement said "|x|+x = 2x", that statement would not be sufficient because we are unable to isolate a unique numerical value (if x=2 then |x|+x=4, but if x=1 then |x|+x=2)

Here is how I would approach the question:
If x is an integer, what is the value of x^2 + x ?

(1) x^2 = |x|

(2) x^2 + x > 0


Statement 1
For almost all integers, x^2 will be bigger than |x|. The only integers for which x^2=|x| are -1, 0, and 1 (visualize the absolute value "V" graph, and the x^2 parabola graph on a plot; if you know what the graphs look like you will realize right away that there are just 3 points of intersection). So this statement tells us that x = {-1, 0 or 1}. x=-1 would make x^2+x = 2, but x=0 or -1 would make x^2+x = 0. INSUFFICIENT

Statement 2
INSUFFICIENT since we could get lots of results for |x|+x depending on what we make x.

Together
Statement 1 limited us to only 2 possible values for x^2+x (2 or 0). Statement 2 tells us that x^2+x > 0, so together the statements guarantee that x^2+x = 2. SUFFICIENT

-Patrick
Last edited by Patrick_GMATFix on Fri Feb 14, 2014 1:30 am, edited 2 times in total.
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by [email protected] » Fri Feb 14, 2014 1:07 am
Hi Sri,

What you've written is incomplete, so I'm not sure if you truly have enough info to prove the correct answer. Here's another way of looking at this question:

We're told that X is an integer. We're asked for the value of X^2 + X?

Fact 1: X^2 = |X|

This is a remarkably restrictive piece of information. The only possible values of X are -1, 0 and 1
If X = -1, then the answer to the question is 0
If X = 0, then the answer to the question is 0
If X = 1, then the answer to the question is 2
Fact 1 is INSUFFICIENT

Fact 2: X^2 + X > 0

There are LOTS of values that fit this Fact.
If X = 1, then the answer to the question is 2
If X = 2, then the answer to the question is 6
Fact 2 is INSUFFICIENT

Combined, we have only 3 possible values from Fact 1; -1 and 0 do NOT fit with Fact 2, so the only possibility is X = 1
Combined, SUFFICIENT

Final Answer: C

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by GMATGuruNY » Fri Feb 14, 2014 3:02 am
The solutions above are great.
I do not recommend using algebra to evaluate statement 1.
But if really insist, here's one approach:

x² = |x|
|x||x| = |x|
|x||x| - |x| = 0
|x| (|x| - 1) = 0.

|x| = 0 when x=0.
|x| - 1 = 0 when x=-1 or x=1.
Thus, valid solutions for statement 1 are x=-1, x=0, x=1.
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