Is X > 1 ? GMAT Data Sufficiency

Question: Is X > 1 ?

1/x > 1
x > 0

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

Correct Answer: A

Solution and Explanation
Approach Solution 1:

We can be confident that x is positive because 1/x > 1 (since the inequality is never true if x is negative). So we can safely multiply by x on both sides without inverting the inequality to find that 1 > x. So Statement 1 is certainly sufficient on its own.
Explaining further

In Statement 1 it states

\(\frac{1}{x}>1 \)

If we multiply x to both sides we will get

\(\frac{1}{x}*x>1*x \)

\(\frac{x}{x}>1x \)

x > 1

Since we are aware that x can never be greater than 1
We will get
x<1
in Statement 2 it states
x>0
We can take x to be a fraction or any number greater than 1.

Let's say that fraction is \(\frac{1}{2},\frac{1}{3} \)

x>0 does not indicate that the value must be higher than one.
Therefore
Yes, X > 1

Approach Solution 2:
1) |x + 1| = 2|x - 1|

two possibilities:
a) x+1=2(x-1) or x=3 (consider x+1 and x-1 to be of same sign)
b) x+1=2(1-x) or x=1/3 (consider x+1 and x-1 to be of different signs)
not sufficient, x may or may not be between -1 and 1

2) |x - 3| ≠ 0
=> x≠3
not sufficient, x may or may not be between -1 and 1

Together, x≠3 so x is 1/3 which is between -1 and 1
sufficient

Approach Solution 3:
For statement 1, we need to actually break it down to 2 separate statements to account for both scenarios:

Option #1 ->
x + 1 = 2(x-1)
x + 1 = 2x -2
x = 2x -3
3 + x = 2x
3 = x {or also Option #2}

Option #2 -> x + 1 = -2(x-1)
x + 1 = -2x +2
x = -2x + 1
3x = 1
x = 1/3

Statement (1) is insufficient because |x| could be 3 or 1/3. So the answer is sometimes yes, sometimes no which means insufficient.

Statement (2) is insufficient because we are told that |x -3| ≠ 0. This means that as long as x ≠ 3, then we're ok. There are far too many options so we certainly have a sometimes yes, and sometimes no.

But together, from Statement (1) we have either 3 or 1/3 and from statement (2) we have everything BUT 3. So the only overlap we have for possible values to make both statements true is 1/3. So the answer is Yes, |x| IS less than 1 and we have enough info to answer the question.

“Is X > 1 ?”- is a topic of thr GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide". 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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Is X > 1 ? GMAT Data Sufficiency

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