If x 0, is |x| > 1 (1) x/|x| < x (2) (x^2)^(1/2) = x

@bunnuel

Why is 1) not sufficient, as |x|>0 then x<x^2 and therefore x<-1 or x>1 which is sufficient to answer the question. What am i missing?

If you try -1/2 as a value, it will also satisfy the inequality in Statement 1 but |x| is not greater than 1

P.S. For modulus questions, its always safe to try 4 different values because it gives us the entire range to check from
Integer > 1
Integer < -1
Fraction between 0 and 1
Fraction between 0 and -1

Posted from my mobile device

Statement 1: \(\frac{x}{|x|}<x\)
CRITICAL POINTS occur when the two sides of an inequality are EQUAL or when the inequality is UNDEFINED.
\(\frac{x}{|x|}=x\) when x=1 or x=-1
\(\frac{x}{|x|}<x\) is undefined when x=0

The critical points are x=-1, x=0 and x=1, implying the following number line:
..........-1..........0..........1..........
To determine which ranges for x are valid, test one value to the left and one value to the right of each critical point.
If we test x=-2, x=-1/2, x=1/2 and x=2, only x=-1/2 and x=2 satisfy \(\frac{x}{|x|}<x\), implying that the valid ranges are -1<x<0 and x>1.

Case 1: -1<x<0
In this case |x|<1, so the answer to the question stem is NO.
Case 2: x>1
In this case |x|>1, so the answer to the question stem is YES.
INSUFFICIENT.

A useful property to know:
\(\sqrt{{x^2}}=|x|\)

Statement 2, rephrased: |x| = x
Since the prompt states that x is nonzero, x must be POSITIVE.
If x=1, then |x|=1, so the answer to the question stem is NO.
If x=2, then |x|>1, so the answer to the question stem is YES.
INSUFFICIENT.

Statements combined:
Only Case 2 satisfies both statements.
In Case 2, the answer to the question stem is YES.
SUFFICIENT.

Hi Bunuel, chetan2u Kinshook
For the second statement, cant the analysis be such-
Say x^2= 4, x=+ or - 2?
The square root of a number can be negative, right?
Please let me know what are the gaps in my understanding.
Using this logic I chose E.

No, square root is always positive.
\(\sqrt{4}\)=2
\(x^2=4\) means x can be 2 or -2