Is the value of x less than 0?

Is x<0?

(1) The product of x and y is negative.
xy is negative. So one of x or y is negative and other positive.
If y>0, x<0, but if y<0, x>0.

(2) The difference between x and its reciprocal is positive.
\(x-\frac{1}{x}>0...\frac{x^2-1}{x}>0\)
If \(x>0..x^2-1>0..x>1\)....\(x=2....2-\frac{1}{2}=\frac{3}{2}\)....x>0
If \(x<0..x^2-1<0....x<0\)...\(x=-\frac{1}{2}...-\frac{1}{2}-(-2)=\frac{3}{2}\)....x<0

Combined..
x<0 or x>1
Insuff

E
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We are to determine if x is less than 0.

Statement 1: xy<0
Clearly insufficient. This is because x can be either positive or negative depending on whether y is positive or negative. When y is positive, x is negative, and when y is negative, x is positive. The answer is, therefore, yes and no.

Statement 2: The difference between x and it's reciprocal is positive.
Statement 2 is not sufficient. This is because x can be 51, implying its reciprocal is 15. 51-15=36, leading to answer No to the question posed. X can also be -15, and its reciprocal is -51. -15--51=36 satisfying the given condition and this now leads to the answer Yes to the question posed.

1+2
Still insufficient. Both statements do not restrict the value of x to less than 0, equal to 0, or greater than 0. When y is negative, x=51 satisfies statement 1 and statement 2, and leads to the answer No to the question asked.
On the other hand, when y>0 x can equal -15, which satisfies both statements and leads to the answer Yes.

The right answer in my view is E.

(1) The product of x and y is negative.
--> x & y have opposite signs (If x > 0, y < 0 or of x <0, y > 0) --> Insufficient

(2) The difference between x and its reciprocal is positive.
--> x - 1/x > 0
--> (x^2 - 1)/x > 0

Formula: If A/B > 0 --> Either both A>0 & B>0 or both A<0 & B<0
Similarly, if (x^2 - 1)/x > 0

Case 1: x^2 - 1 > 0 & x > 0
--> x > 1 (x is positive)

Case 2: x^2 - 1 < 0 & x < 0
--> -1 < x < 0 (x is negative)
--> Insufficient

Combining (1) & (2),

Also, x can be either positive or negative --> Insufficient

IMO Option E

Is the value of x less than 0?

Is x < 0?

(1) The product of x and y is negative.
xy < 0
x < 0 or y < 0
insufficient

(2) The difference between x and its reciprocal is positive.

x - 1/x > 0
x2 -1 > 0
(x+1)(x-1) >0
x< -1 or x> 1 - we get that x can be both positive or negative

insufficient

taken (1) + (2) together, x can still be both positive and negative - insufficient
Thus, E is the correct answer

We have to find whether or not x<0.

(1) xy<0
=> Either of x and y is negative and the other is positive.
So, we do not have sufficient information to find out whether or not x<0.
Thus, insufficient.

(2) x-\(\frac{1}{x}\)<0
=> \(\frac{(x^2-1)}{x}\)<0
=> x>1 or -1<x<0
So, x can be positive or negative.
Thus, insufficient.

From (1) and (2) together, we do not have sufficient information to find out whether or not x<0.
Thus, insufficient.

Therefore, the correct answer is option E.

Imo. E

Is the value of x less than 0?
Is x = -ve?

(1) The product of x and y is negative.
xy<0, x can or can not be -ve. Insufficient.

(2) The difference between x and its reciprocal is positive.
x-1/x = +ve
x can be 2 or -1/2, So x can be or can not be -ve. Insufficient.

1 + 2)
X can be or can not be -ve. Insufficient.
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(1) The product of x and y is negative. insufic.

\(xy<0:[x>0,y<0]…or…[x<0,y>0]\)

(2) The difference between x and its reciprocal is positive. insufic.

\(x-\frac{1}{x}>0…\frac{x^2-1}{x}>0…\frac{(x-1)(x+1}{x}>0…[x<-1,x>1]\)

(1 & 2): insufic.

Answer (E)

Is the value of x less than 0?

is X negative?

(1) The product of x and y is negative.
meaning either x or y is negative.
we don't know which one.
Therefore, insuffucient

(2) The difference between x and its reciprocal is positive.
x- (1/x) is positive
(1/x)-x is positive
there is no such number that will give those both statement true
insufficient

together, we know y doesn't give anything and with those statement in b doesn't add any value.

Therefore, E

The trap here is to blindly assume value of X to be an integer.

Statement 1 is clearly not sufficient as X can be greater than 0 or less than 0. Not sufficient.

Statement 2-.. tricky part.

Let's see , X -1/X >0
(X^2-1)/X > 0, at first one might think for this expression to be true, X has to be positive.. and if one thinks that way, he/she has made a wrong assumption of X to be a positive integer..

Take X = 4
(16-1)/4 > 0, I get a no
Take X = -0.5
(0.25-1)/-0.5 > 0, but X is less than 0 and I get a yes . This statement is not sufficient as well .

Combining both statements is of no use because X can be positive or X can be negative.. eliminate C and E is the right answer..

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