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# If x is an integer, what is the value of x?

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Math Expert
Joined: 02 Sep 2009
Posts: 55609
Re: If x is an integer, what is the value of x?  [#permalink]

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10 Sep 2014, 07:49
1
1
NikhilDev wrote:
Hey bunuel,
I got x=-1,2 from the first statement.
But in the second statement, i took positive and negative possibilities and came up with 4 equations.
the 4 equations are
1.(x^2)-x=2
2.(x^2)+x=2
3. -(x^2)+x=2
4. -(x^2)-x=2
Out of these four equations, according to me, only the 1st and the second have real solutions.
So from the 1st equation above i got x=-1,2(Same as the result from our First Statement )
and from the second equation i got x=1,-2.
But everywhere people have written that they got x=-2,2 from the second statement. Can you please explain it to me as to what the issue is.

Does -1, or 1 satisfy the equation? No. So, x cannot be -1 or 1.

Next, you get x^2 - x = 2 for positive x, so when solving you should discard negative solutions. Similarly, you get x^2 + x = 2 for negative x, so when solving you should discard positive solutions.
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Re: If x is an integer, what is the value of x?  [#permalink]

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04 Feb 2016, 07:05
Bunuel wrote:
rohitgoel15 wrote:
If x is an integer, what is the value of x?
1) |x - |x2|| = 2
2) |x2 - |x|| = 2

I saw the solution and I think i cant even get close. On the test, I would prefer not to solve this question. But is there a short way to make an educated guess.

Answer is not E as given in above posts, it's C. Also note that 1 and -1 does not satisfy statement (2).

If x is an integer, what is the value of x?

(1) |x - |x^2|| = 2. First of all: $$|x^2|=x^2$$ (as $$x^2$$ is a non-negative value). Square both sides: $$(x-x^2)^2=4$$ --> factor out $$x$$: $$x^2*(1-x)^2=4$$ --> as $$x$$ is an integer then $$x=2$$ or $$x=-1$$ (by trial and error: the product of two perfect square is 4: 1*4=4 or 4*1=4). Not sufficient.

(2) |x^2 - |x|| = 2 --> square both sides: $$(x^2-|x|)^2=4$$ --> factor out $$|x|$$: $$x^2*(|x|-1)^2=4$$ --> as $$x$$ is an integer then $$x=2$$ or $$x=-2$$. Not sufficient.

(1)+(2) Intersection of the values from (1) and (2) is $$x=2$$. Sufficient.

Hope it's clear.

if x^2 = 4 then isn't x= (2,-2) ?
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Joined: 02 Sep 2009
Posts: 55609
Re: If x is an integer, what is the value of x?  [#permalink]

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04 Feb 2016, 07:10
1
email2vm wrote:
Bunuel wrote:
rohitgoel15 wrote:
If x is an integer, what is the value of x?
1) |x - |x2|| = 2
2) |x2 - |x|| = 2

I saw the solution and I think i cant even get close. On the test, I would prefer not to solve this question. But is there a short way to make an educated guess.

Answer is not E as given in above posts, it's C. Also note that 1 and -1 does not satisfy statement (2).

If x is an integer, what is the value of x?

(1) |x - |x^2|| = 2. First of all: $$|x^2|=x^2$$ (as $$x^2$$ is a non-negative value). Square both sides: $$(x-x^2)^2=4$$ --> factor out $$x$$: $$x^2*(1-x)^2=4$$ --> as $$x$$ is an integer then $$x=2$$ or $$x=-1$$ (by trial and error: the product of two perfect square is 4: 1*4=4 or 4*1=4). Not sufficient.

(2) |x^2 - |x|| = 2 --> square both sides: $$(x^2-|x|)^2=4$$ --> factor out $$|x|$$: $$x^2*(|x|-1)^2=4$$ --> as $$x$$ is an integer then $$x=2$$ or $$x=-2$$. Not sufficient.

(1)+(2) Intersection of the values from (1) and (2) is $$x=2$$. Sufficient.

Hope it's clear.

if x^2 = 4 then isn't x= (2,-2) ?

Yes, but if x=-2, then x^2*(1-x)^2 does not equal to 4.
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Re: If x is an integer, what is the value of x?  [#permalink]

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04 Feb 2016, 07:52
NikhilDev wrote:
Hey bunuel,
I got x=-1,2 from the first statement.
But in the second statement, i took positive and negative possibilities and came up with 4 equations.
the 4 equations are
1.(x^2)-x=2
2.(x^2)+x=2
3. -(x^2)+x=2
4. -(x^2)-x=2
Out of these four equations, according to me, only the 1st and the second have real solutions.
So from the 1st equation above i got x=-1,2(Same as the result from our First Statement )
and from the second equation i got x=1,-2.
But everywhere people have written that they got x=-2,2 from the second statement. Can you please explain it to me as to what the issue is.

You are making a mistake while interpreting different scenarios after you open up the absolute sign.

Per statement 2, |x^2-|x||=2

Consider 2 cases,

Case 1: when $$x \geq 0$$, $$|x| =x$$ --->$$|x^2-|x||=2$$ becomes $$|x^2-x|=2$$ ---> $$x^2-x=\pm 2$$ , giving you 2 quadratic equations

$$x^2-x-2=0$$ and $$x^2-x+2=0$$ (no real solutions, eliminate). From $$x^2-x-2=0$$ , you get x=2 and x=-1 but x=-1 is NOT allowed as you are assuming that for this case, $$x \geq 0$$.

Thus x=2 is the only possible solution from this scenario.

Case 2: when $$x<0$$---> $$|x| = -x$$ ---> $$|x^2-|x||=2$$ becomes $$|x^2+x|=2$$---> $$x^2+x=\pm 2$$ , giving you 2 quadratic equations again,
$$x^2+x-2=0$$ and$$x^2+x+2=0$$ (no real solutions, eliminate). From $$x^2+x-2=0$$, you get x=-2 and x=1 but x=1 is NOT allowed as you are assuming that for this case, $$x < 0$$.

Thus x=-2 is the only possible solution from this scenario.

Finally, when you combine both the scenarios, you see that the allowed values are $$x=\pm 2$$. Thus this statement is NOT sufficient to answer the question asked.

Hope this help. You have to be very careful about your assumptions in absolute value questions.
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Re: If x is an integer, what is the value of x?  [#permalink]

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04 Feb 2016, 09:46
1
rohitgoel15 wrote:
If x is an integer, what is the value of x?

(1) |x - |x^2|| = 2
(2) |x^2 - |x|| = 2

Question : x=?

Statement 1: |x - |x^2|| = 2
The solutions of the given equation are
x=2 and -1
NOT SUFFICIENT

Statement 2: |x^2 - |x|| = 2
The solutions of the given equation are
x=2 and -2
NOT SUFFICIENT

Combining the two statements
x = 2
SUFFICIENT

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Re: If x is an integer, what is the value of x?  [#permalink]

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04 Feb 2016, 18:31
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If x is an integer, what is the value of x?

(1) |x - |x^2|| = 2
(2) |x^2 - |x|| = 2

In the original condition, there is 1 variable(x), which should match with the number of equations. So you need 1 equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.
For 1), it becomes |x^2|=x^2 -> x-x^2=-2, +2. In x-x^2=-2, +2, x^2-x-2=0, (x-2)(x+1)=0 -> x=2, -1, which is not unique and not sufficient.
For 2), in x^2-|x|=-2, 2, x^2=|x|^2 is derived. |x|^2-|x|=2, |x|^2-|x|-2=0, (|x|-2)(|x|+1)=0 -> x|=2, -1. -1 is impossible. |x|=2 -> x=-2,2, which is not unique and not sufficient.
When 1) & 2), x=2 is unique, which is sufficient.

-> For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Re: If x is an integer, what is the value of x?  [#permalink]

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07 May 2016, 11:43
A slightly different approach:

Statement 1
X is -1 or 2
Insufficient

Statement 2
X is 2 or -2
Insufficient

Combined
2 is the only common root for both equations = hence X must be 2.
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Re: If x is an integer, what is the value of x?  [#permalink]

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09 Dec 2018, 09:44
Bunuel wrote:
rohitgoel15 wrote:
If x is an integer, what is the value of x?
1) |x - |x2|| = 2
2) |x2 - |x|| = 2

I saw the solution and I think i cant even get close. On the test, I would prefer not to solve this question. But is there a short way to make an educated guess.

Answer is not E as given in above posts, it's C. Also note that 1 and -1 do not satisfy statement (2).

If x is an integer, what is the value of x?

(1) |x - |x^2|| = 2. First of all: $$|x^2|=x^2$$ (as $$x^2$$ is a non-negative value). Square both sides: $$(x-x^2)^2=4$$ --> factor out $$x$$: $$x^2*(1-x)^2=4$$ --> as $$x$$ is an integer then $$x=2$$ or $$x=-1$$ (by trial and error: the product of two perfect square is 4: 1*4=4 or 4*1=4). Not sufficient.

(2) |x^2 - |x|| = 2 --> square both sides: $$(x^2-|x|)^2=4$$ --> factor out $$|x|$$: $$x^2*(|x|-1)^2=4$$ --> as $$x$$ is an integer then $$x=2$$ or $$x=-2$$. Not sufficient.

(1)+(2) Intersection of the values from (1) and (2) is $$x=2$$. Sufficient.

Hope it's clear.

Do square and mod cancel each other?
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If x is an integer, what is the value of x?  [#permalink]

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03 Feb 2019, 04:16
In this circumstance, especially in the test, doing pure algebra may not be your first instinct. By looking at the question, we know that x is an integer and from the statements, it is clear we are dealing with small numbers so draw a table.

Attachment:

4.PNG [ 5.42 KiB | Viewed 307 times ]

Statement 1 alone: both x = -1 and x = 2 gives 2 --> Not sufficient
Statement 2 alone: both x = -2 and x = 2 gives 2 --> Not sufficient

Together, when they overlap, only x = 2 gives 2 - Sufficient.

The table seems tedious but squaring 1's and 2's should come to you instantly and this table can be drawn in <1.5 mins and hence this question can be solved in <2 mins.
Attachments

4.PNG [ 6.93 KiB | Viewed 304 times ]

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Re: If x is an integer, what is the value of x?  [#permalink]

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06 Feb 2019, 09:18
This is the correct explanation. Correct answer is E.

Apex231 wrote:
If x is an integer, what is the value of x?
1) |x - |x^2|| = 2
x^2 is always positive. so |x^2| = x^2

x - x^2 is negative since X^2 > x for an integer

-(x - x^2) = 2
-x + x^2 = 2
x^2 -x - 2 = 0
x^2 -2x +x - 2 = 0
(x-2) (x+1)
x = 2 or x = -1 , two values , not sufficient.

2) |x2 - |x|| = 2
x^2 - |x| is positive since X^2 > x for an integer

|x| can be positive or negative. so two scenarios.

x^2 -x = 2
x^2 -x - 2 = 0
(x-2)(x+1) = 0

or

x^2 + x = 2
x^2 + x - 2 = 0
(x+2) (x-1) = 0

Multiple values so not sufficient.

(1) + (2) , still not sufficient.

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Re: If x is an integer, what is the value of x?   [#permalink] 06 Feb 2019, 09:18

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