Is |x| < 1? GMAT Data Sufficiency

Question: Is |x| < 1?

1. |x + 1| = 2|x - 1|
2. |x - 3| ≠ 0

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

Correct Answer: C 

Solution and Explanation:
Approach Solution 1:
It has asked in the question to find if the value of |x| < 1.
And we are given two cases.
(1) |x + 1| = 2|x - 1|
(2) |x - 3| ≠ 0
We have to check that whether those cases are sufficient to solve the problem or not.
Firstly it is important to understand what is asked in the question.
It has asked if |x| < 1
The modulus of x is less than 1 means x is a number lying between -1 and 1 so that after taking the modulus the value should be less than 1.
So we should find out whether -1 < x < 1

Statement 1 indicates that |x + 1| = 2|x - 1|

Here the points where modulus values are zeroes are key breaking points.
x+1 = 0
Or, x = -1
Also, x-1 = 0
Or, x = 1.
So x = -1,1 are the breaking points in this equation. Therefore we have to check the answer in all three ranges which are –
x < -1
-1 < x < 1
x > 1

So let us take 1st range when x < -1
Or, x + 1 < 0
And x - 1 < 0
Therefore | x+ 1 | = - (x + 1)
And | x - 1| = - (x - 1)

Putting these in the first equation we get,
|x + 1| = 2|x - 1|
Or, -(x+1) = 2 * (- (x-1))
Or, - x - 1 = -2x + 2
Or, 2x - x = 1 + 2
Or, x = 3

This value of x is greater than zero and it is absurd because we assumed x < -1

Now let's take -1 < x < 1
Now x + 1 > 0
And x - 1 < 0
|x + 1| = 2|x - 1|
Or, (x+1) = 2 * (- (x-1))
Or, x + 1 = -2x + 2
Or, 2x + x = 2 - 1
Or, 3x = 1
Or, x = \(\frac{1}{3}\)

This value of x is correct because we assumed -1 < x < 1

Now for x > 1
Now x + 1 >0
And x - 1 > 0
|x + 1| = 2|x - 1|
Or, (x+1) = 2 * (x-1)
Or, x + 1 = 2x - 2
Or, 2x - x = 2 + 1
Or, x = 3
This value of x is correct because we assumed x > 1
So we got two values of x which are \(\frac{1}{3}\) and 3 which is correct. However, the value x = 3 is out of the range of (-1, 1).

Therefore, statement 1 alone is not sufficient to find the answer.

Statement 2 indicates that, |x - 3| ≠ 0
It is given that |x-3| is not equal to 0
Which implies x- 3 ≠ 0
Or, x ≠ 3

But this condition doesn’t say whether the values of x lie between -1 to 1 or not.
Therefore Statement 2 is also not sufficient to find the answer.

Now combining both statements,
From statement 1, we get x = 3 and \(\frac{1}{3}\)
From statement 2, we get x ≠ 3

Hence, the only value of x that remained is x = \(\frac{1}{3}\)
So we can say that -1 < x < 1
Therefore both statements are sufficient to find out the answer.

Approach Solution 2:
The problem statement asks that if the value of |x| < 1.

|x|<1 is satisfied by the range -1<x<1.

In other words, the question is asking us to find out if -1<x<1.
It is required to remember that |x| = √(x^2 ).

Statement 1 alone: |x + 1| = 2|x - 1|

We can say, |x+1| = √(x+1)^2 and |x-1| = √(x−1)^2
Therefore, √(x+1)^2 = 2 √(x−1)^2.
On squaring both sides and simplifying, we get,
3x^2– 10 x + 3 = 0.
Solving this quadratic equation, we get x = 3 or x = 1/3.

This is true. The distance of the number 3 from -1 is indeed double that of its distance from 1.

Likewise, the distance of the number 1/3 from -1 is double that of its distance from 1.
Therefore, it actually ties in with the definition of |x-a| on which statement 1 is framed.

However, all this does not give us a unique value of x.
Therefore, Statement 1 is insufficient.

Statement 2 alone: |x - 3| ≠ 0
It implies that x is not equal to 3.

Therefore, statement 2 is clearly insufficient to determine if -1<x<1. 

Combining both statements, we get:
x has to be equal to 1/3 since it cannot be equal to 3.

Therefore, both statements together are sufficient to find the answer.

Approach Solution 3:
The problem statement asks if the value of |x| < 1.
This implies that if the value of  x is between -1 and 1

Statement 1: |x+1| = 2|x-1|
If x < -1
The equation will yield x = 3 which is clearly wrong. It simply cannot be since x should be less than -1.
Therefore, we can not determine the solution if x < -1
If -1 < x < 1
The equation will give 1/3 as a solution.
If x > 1
The equation will give 3 as a solution.
Since the values of x may be outside of (-1,1).
Therefore, statement 1 alone is INSUFFICIENT.

Statement 2 : |x - 3| ≠ 0
It implies that x is not equal to 3.
Therefore x does not lie between -1 and 1. 
Hence, statement 2 alone is INSUFFICIENT

Combining both statements 1 and 2  we get:
Statement 2 removes 3 as a possible root.

Therefore both statements together are SUFFICIENT.

“Is |x| < 1”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Prep Plus”. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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