Marcab wrote:

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

(A) \(x>1\)

(B) \(x>-1\)

(C) \(|x|<1\)

(D) \(|x|>1\)

(E) \(-1<x<0\)

We can simplify the given inequality:

x/|x| < x

x < (x)|x|

(x)|x| > x

If x is positive, we can divide both sides by x and obtain |x| > 1.

If x is negative, we can also divide both sides by x, but we have to switch the inequality sign, so we have |x| < 1.

We see that if x is positive, |x| > 1, which is choice C, and if x is negative, |x| < 1, which is choice D. However, since we don’t know whether x is positive or negative, both choice C and choice D “can be true,” not “must be true.”

Let’s analyze further. If x is positive, |x| = x. So, |x| > 1 means x > 1, which is choice A. If x is negative, |x| = -x. So, |x| < 1 means -x < 1 or x > -1. However, because x is negative, we have -1 < x < 0, which is choice E. Again, since we don’t know whether x is positive or negative, both choice A and choice E “can be true,” not “must be true.”

This leaves choice B as the correct answer. In fact, it’s the correct choice because the inequality x > -1 includes both x > 1 and -1 < x < 0. So, regardless of whether x is positive or negative, we can say x > -1.

Answer: B

_________________

Scott Woodbury-Stewart

Founder and CEO

GMAT Quant Self-Study Course

500+ lessons 3000+ practice problems 800+ HD solutions