Is |x - 1| < 1? 1. \((x-1)^2\leq1\)  2. \(x^2-1 > 0\) GMAT Data Sufficiency

Question: Is |x - 1| < 1?

1. \((x-1)^2\leq1\) 
2. \(x^2-1 > 0\)

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.

“Is |x - 1| < 1?” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken f0rom the book "GMAT Quantitative Review". 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.

Answer:

Approach Solution 1:

Considering S1, we have \((x-1)^2\leq1\)

Since both the sides of the inequality are non-negative then we can take square root from both parts: |x – 1| \(\leq\)1, so |x – 1| can be less than 1 (answer YES), as well as equal to 1, for x = 2 or x = 0 (answer NO)

Hence, this statement is Not Sufficient.
Notice that |x – 1| \(\leq\)1, means 0 x\(\leq\) 2
Now consider S2, we have \(x^2-1 > 0\)

Rearrange \(x^2>1\)

Again, since both sides of the inequality are non-negative then we can take square root from both parts: |x| > 1.
If x = 1.5 then the answer is YES but if x = 2 then the answer is NO.

Hence, this statement is Not Sufficient.
(1) + (2) x = 1.5 and x = 2 satisfy both the statements and give different answers to the questions.

Hence, this statement is Not Sufficient.

Correct Answer: E

Approach Solution 2:

There can be two scenarios for the main statement

1. a) When x – 1 < 0, x < 1

Here, - x + 1 < 1

-x < 0
x > 0
0 < x < 1= x value should be in the range

b) x – 1 > 0; x > 1
x – 1 < 1
x < 2
1 < x < 2 = another range of x value
Now let’s look at the two options

1)\((x-1)^2\leq1\)
(x – 1)\(\leq\) 1; x – 1\(\leq\) -1
x \(\leq\)2 and x\(\leq\) -2

Doesn’t satisfy the range
Hence, this statement is Not Sufficient.

2)\(x^2-1 > 0\)

\(x^2>1\)

It can take any positive value and doesn’t satisfy the range.

Correct Answer: E

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