M09-22

Official Solution:

If \(x \ne 0\) and \(\frac{x}{|x|} \lt x\), which of the following must be true?

A. \(x \gt 1\)
B. \(x \gt -1\)
C. \(|x| \lt 1\)
D. \(|x| > 1\)
E. \(-1 \lt x \lt 0\)


To begin with, note that the problem is asking to identify which option MUST be true, not just which one COULD be true. Next, let's determine the range(s) for \(x\), given that \(x\) is not equal to 0 and \(\frac{x}{|x|} \lt x\). Consider two cases:

Case 1: If \(x \lt 0\), then \(|x|=-x\). In this case, we have \(\frac{x}{-x} \lt x\), which simplifies to \(-1 \lt x\). Since we are considering the range where \(x \lt 0\), then for this case we have \(-1 \lt x \lt 0\).

Case 2: If \(x \gt 0\), then \(|x|=x\). In this case, we have \(\frac{x}{x} \lt x\), which simplifies to \(1 \lt x\).

Thus, \(\frac{x}{|x|} \lt x\) implies that \(-1 \lt x \lt 0\) or \(x \gt 1\). The only option that is always true is B, which states that \(x \gt -1\). This is because ANY \(x\) from the range \(-1 \lt x \lt 0\) or \(x \gt 1\) will definitely be greater than \(-1\). Thus, B, which states that \(x\) is greater than -1, must be true!

Option A, \(x \gt 1\), is not necessarily true since \(x\) could be -0.5.

Option C, \(|x| \lt 1\), is not necessarily true since \(x\) could be 2.

Option D, \(|x| \gt 1\), is not necessarily true since \(x\) could be -0.5.

Option E, \(-1 \lt x \lt 0\), is not necessarily true since \(x\) could be 2.


Answer: B