Official Solution:


If \(x ≠ 0\), is \(\frac{|x|}{x} < 1\) ?

When \(x < 0\), then \(|x|=-x\), and in this case \(\frac{|x|}{x}=\frac{-x}{x} = -1\). So, when \(x < 0\), the answer to the question is always YES.

When \(x > 0\), then \(|x|=x\), and in this case \(\frac{|x|}{x}=\frac{x}{x} = 1\). So, when \(x > 0\), the answer to the question is always NO.

(1) \(x < 1\). Not sufficient.

(2) \(x > −1\). Not sufficient.

(1)+(2) We get \(-1 < x < 1\). We still cannot say whether \(x\) is negative or positive. Not sufficient.


Answer: E
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E because
Modx/x is 1 for all positives and -1 for all negatives.

Posted from my mobile device

Statement 1: x<1
x can be any value from -1 and below as x cannot be zero
With the given condition, |x|/x<1,
|-1|/-1= -1 <1 statement is true
|-10|/-1=-10<1, statement is true

Statement shall remain true for any value of x<1.

Hence 1 is sufficient

Statement 2: X>-1
X can be any value from 1

|1|/1= 1 , so the equation |x|/x<1, is not satisfied hence answer is false
|10|/10=1,the equation |x|/x<1, is not satisfied hence answer is false.
For any value of x, answer is false.
Hence statement 2 is sufficient.

Answer is D

Please check as x can be fraction also....let x be -.1
Then |x| will be .1
|X|/x will be less than 1(-ve in this case)...however when x is positive function will be always 1....so no answer from 2 while 1 nullify as function will be always 1...so correct choice seems to be E

Posted from my mobile device

Let's test some values

(1) x < 1

x=1/2.............Answer is No

x=-1/2.............Answer is Yes

Insufficient

2) x > −1

x=1/2.............Answer is No

x=-1/2.............Answer is Yes

Insufficient

combining 1 & 2

-1 < x <1..........We can use values above

Insufficient

Answer: E

Question: is |x|/x < 1?

|x|/x can only be either +1 (if x is positive) or -1 (if x is Negative)
For |x|/x< 1, x must be negative

Question REPHRASED: is x < 0?

Statement 1: x < 1

NOT SUFFICIENT

Statement 2: x > -1

NOT SUFFICIENT

COmbining the statements

-1 < x < 1

Still x may be positive as well as Negative hence

NOT SUFFICIENT

Answer: Option E
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Solution

Step 1: Analyse Question Stem

• \(x ≠ 0\)
• We need to find if \(\frac{|x|}{x} < 1\)

o Case 1: If \(x < 0\), then \(\frac{|x|}{x }= \frac{-x}{x }= -1 < 1\)
o Case 2: If \(x > 0\) then \(\frac{|x|}{x} = \frac{x}{x} = 1 \)

So, to find if \(\frac{|x|}{x} < 1\) we actually need to find if x < 0.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE

Statement 1: \(x < 1\)

• We cannot be sure from this statement that x < 0. For example, x can be 0.5 or it can be -0.5

Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.
Statement 2: \(x > -1\)

• Here also, we cannot be sure from this statement that x < 0. For example, x can be 0.5 or it can be -0.5

Hence, statement 2 is also NOT sufficient and we can eliminate answer Option B.

Step 3: Analyse Statements by combining.

• From statement 1:\( x < 1\)
• From statement 2: \(x > -1\)
• On combining both statements, we get, \(-1 < x < 1\)

o Still we cannot be sure that x < 0. For example, x can be 0.5 or it can be -0.5

Thus, the correct answer is Option E.
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