If x 0, what is the value of x?

Official Solution:


If \(x ≠ 0\), 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 \(-x^2\)

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. 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 \(x ≠ 0\), 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.


Answer: E