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

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

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

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


Need hep .How to approach such problems in an efficient way. $$$$ $$$$ $$$$

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prab.sahi06 wrote:
If x is an integer, what is the value of x?

(1) |x - |x^2|| = 2
(2) |x^2 - |x|| = 2
In statement 1, |x²| is redundant: since x² cannot be negative, |x²| = x².

Statement 1: |x-x²|=2
x - x² = ±2
x(1-x) = ±2.

The resulting equation implies that x is a factor of ±2, yielding the following options:
x=±1 or x=±2.
Check which of these values are valid solutions for |x-x²| = 2.

If x=1, then |x-x²| = |1 - 1²| = 0.
If x=-1, then |x-x²| = |-1 - (-1)²| = 2.
If x=2, then |x-x²| = |2 - 2²| = 2.
If x=-2, then |x-x²| = |-2 - (-2)²| = 6.

Since it's possible that x=-1 or that x=2, INSUFFICIENT.

Statement 2: |x² -|x||=2

x²-|x| = ±2
Since x² = |x|*|x|, we can factor out |x|:
|x| (|x|-1) = ±2.

The resulting equation implies that |x| is a factor of ±2, yielding the following options:
|x|=1 or |x|=2, with the result that x=±1 or x=±2.
Check which of these values are valid solutions for |x² -|x||=2.

If x=-1, then |x² -|x|| = |(-1)² - |-1|| = 0.
If x=1, then |x² -|x|| = |1² - |1|| = 0.
If x=2, then |x² -|x|| = |2² - |2|| = 2.
If x=-2, then |x² -|x|| = |(-2)² - |-2|| = 2.

Since it's possible that x=2 or that x=-2, INSUFFICIENT.

Statements 1 and 2 combined:
Both statements are satisfied only by x=2.
SUFFICIENT.

The correct answer is C.

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GMATGuruNY wrote:
prab.sahi06 wrote:
If x is an integer, what is the value of x?

(1) |x - |x^2|| = 2
(2) |x^2 - |x|| = 2
In statement 1, |x²| is redundant: since x² cannot be negative, |x²| = x².

Statement 1: |x-x²|=2
x - x² = ±2
x(1-x) = ±2.

The resulting equation implies that x is a factor of ±2, yielding the following options:
x=±1 or x=±2.
Check which of these values are valid solutions for |x-x²| = 2.

If x=1, then |x-x²| = |1 - 1²| = 0.
If x=-1, then |x-x²| = |-1 - (-1)²| = 2.
If x=2, then |x-x²| = |2 - 2²| = 2.
If x=-2, then |x-x²| = |-2 - (-2)²| = 6.

Since it's possible that x=-1 or that x=2, INSUFFICIENT.

Statement 2: |x² -|x||=2

x²-|x| = ±2
Since x² = |x|*|x|, we can factor out |x|:
|x| (|x|-1) = ±2.

The resulting equation implies that |x| is a factor of ±2, yielding the following options:
|x|=1 or |x|=2, with the result that x=±1 or x=±2.
Check which of these values are valid solutions for |x² -|x||=2.

If x=-1, then |x² -|x|| = |(-1)² - |-1|| = 0.
If x=1, then |x² -|x|| = |1² - |1|| = 0.
If x=2, then |x² -|x|| = |2² - |2|| = 2.
If x=-2, then |x² -|x|| = |(-2)² - |-2|| = 2.

Since it's possible that x=2 or that x=-2, INSUFFICIENT.

Statements 1 and 2 combined:
Both statements are satisfied only by x=2.
SUFFICIENT.

The correct answer is C.
I was considering so many cases and getting lost.

This is great.Thanks a lot !!

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Without simplification,
Statement 1: |x - |x^2|| = 2
|-1-|(-1)^2||=2 [X=-1]
Or
|2-|(2^2)||=2 [X=2]
We find that X=-1 or X=2; therefore, statement 1 alone is NOT SUFFICIENT.

Statement 2: |x^2 - |x|| = 2
|2^2-|2||=2 [X=2]
Or
|(-2)^2-|-2||=2 [X=-2]

We find that X=2 or X=-2; therefore, statement 2 alone is NOT SUFFICIENT.

Statement 1+2: from statement 1 and 2, we find that X=2. SUFFICIENT.
Answer: C

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