If \(x\neq 0\)  ,What is the Value of x? GMAT Data Sufficiency

Question: If \(x\neq 0\) ,what is the value of x?

1. The distance between x and –x on the number line is equal to the distance between \(x^2\) and \(-x^2\)
2. x ? |x| = 0, where ? represents one of the three operations addition, subtraction, or multiplication

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.

“If ,what is the value of x?”– is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from 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.

Solution and Explanation:

Approach Solution 1:

(1) The distance between x and –x on the number line is equal to the distance between \(x^2\) and \(-x^2\)

The distance between a and b on the number line is |a-b|. Thus, the statement above means |x – (-x)| = | \(x^2\)-(\(-x^2\))|

|2x| = |\(2x^2\)|

|2x| = |2x| * |x|

Reduce by |2x| (note here that we can safely do that since |2x| \(\neq\)0. We are told x 0, thus |2x| \(\neq\)0):

1 = |x|

X can be 1 or -1.
Hence not sufficient.

(2) x ? |x| = 0, where ? represents one of the three operations addition, subtraction or multiplication

If ? is addition we’d have x + |x| = 0, which is true for all non-positive values of x.
If ? is subtraction we’d have x - |x| = 0, which is true for all non-negative values of x.
If ? is multiplication we’d have x * |x| = 0, which is true when x = 0 but since we are told that x \(\neq\)0, then ? cannot be multiplication.

Not sufficient

(1) + (2) If ? is subtraction then x = 1 (from (1)) but if ? is addition then x = -1 (from (1)).
Not sufficient

Correct Answer: E

Approach Solution 2:

(1) x can take 0 or 1 or -1

In all cases the distance between x and –x on the number line is equal to the distance between \(x^2\) and \(-x^2\)

Not sufficient

(2) If x = 0 then ? represents any one of the operations

If x = 0 then ? represents any one of the operations.
If x = -1 then ? represents addition
If x = 1 then ? represents subtraction

Not sufficient

On combining also not sufficient

Correct Answer: E

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