Around the World in 80 Questions (Day 6): If x and y are nonzero

Each alone is sufficient. According to 1. X and Y are negative then their product will be positive and also since we are taking the absolute value into consideration it'd be positive. 2. For X-Y = 0 both should be equal so both would either be positive or negative, either way their product would be positive.
Answer - D

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The Answer is D
Statement 1 says- Both x and y are negative numbers.
If both x and y are negative numbers, then the product xy will be positive, as a negative number multiplied by another negative number results in a positive number.
So |xy| is not equal to -xy. Thus, statement (1) is sufficient to answer the question.

Statement 2 says x-y=0 which basically means that x=y so |xy| = |x * x|
Since the absolute value of a number squared is always positive, so |x * x| = x^2, which is positive.
On the other hand, -xy = -(x * y) = -(x * x), which is the negative of a positive number x^2.
Therefore, |xy| and -xy are not equal. Thus, statement (2) is also sufficient to answer the question.

If x and y are nonzero numbers, does |xy| = -xy ?

(1) Both x and y are negative numbers.
suppose x=-2, y=-2 lxyl=4
-(-2)(-2)=-4 --> 4 does not equal -4 yes
(2) x - y = 0.
x=y --> x=3=y lxyl=9
-xy=-9
so 9 is not equal to -9 yes

Hence D

As per quexstion, x,y are not eaual to 0.
for mode of xy to be -ve, one of the x or y has to be -ve, otherwise (-) will cancel out and xy will become positive, this means each of x & y must have different signs- one +ve and other -ve.

Statement 1:
since both x and y are nevative, (-x)(-y)= xy and it is not equal to -xy.
Sufficient.

Statement 2:
x-y=0, x=y,
both x and y have same signs, hence mode of xy is +ve.
sufficient.

OA: D

given that x & y are non zero numbers
is |xy| = -xy
since LHS will always be +ve so RHS has to be of are of opposite signs to be true of condition
#1
(1) Both x and y are negative numbers.
sufficient to say that RHS is -ve , and we can say NO to |xy| = -xy
sufficient
#2
x - y = 0.
x=y
x & y will be same value and of same sign or opposite sign
condition 1 :
if x = y and are +ve
we get no to condition |xy| = -xy
condition 2 :
if x, y are opposite sign
x=-y
-y-(-y)=0
-y+y=0
we get yes to condition |xy| = -xy

OPTION A is correct

statement 1
If both x and y are positive or both are negative, |xy| = xy.
statement 2
if x-y=0(one if x and y will be positive and other will be negative same no )
If one of x or y is positive, and the other is negative, |xy| = -xy.

Statement 1: Take x = -2 ; y = -3
| -2 * -3| = 6 (not a -ve number)
Eliminated

Statement 2: x = y
Case 1: x = y = -2
|-2 * -2| = 4 (not a -ve number)
Eliminated
Case 2: x = y = 2
|2*2| = 4 (not a -ve number)
Eliminated

Statement 1 & 2 together also don't give any new results.

Correct answer: E

The absolute value of any number is always non-negative. So, |xy| can never be equal to -xy because -xy is negative when xy is positive.

(1) Both x and y are negative numbers.
In this case, xy would be positive, and |xy| would also be positive. So, |xy| = xy, not -xy.
Sufficient

(2) x - y = 0.
This implies x = y. If both x and y are positive or both are negative, |xy| = xy, not -xy. If one is positive and the other is negative, |xy| = -xy, but this contradicts the given that x and y are nonzero numbers.
Sufficient

So, in both cases, |xy| ≠ -xy.
Hence D

For |xy| = -xy to be true. X and Y need to have opposite signs. The value out of the absolute sign will be positive and for -xy to equal a positive number xy itself must be negative, -(-xy) = +xy

1. SUFFICIENT X and Y are of the same sign, meaning |xy| = -xy is not true.
2. SUFFICIENT X and Y are of the same sign. You can plug in numbers. For example, replace Y with 1 and -1.

Postive 1 Negative 1
x-1=0 x-(-1)=0
x=1 x = -1

x and y are both positive. x and y are both negative.

|p| = -p indicates p is negative (or p is zero, but we can ignore zero). We need to find out whether xy is negative.

(1) Both x and y are negative numbers.

If x and y are negative number, the product xy is positive. Hence xy is not negative.

Sufficient.

(2) x - y = 0

This can happen when x and y share the same magnitude and the same sign. We already know that x and y cannot be zero. Hence, x and y are both positive or both negative.

In both the cases, xy is positive. Hence xy is not negative.

Sufficient.

IMO D

-This question basically asks if xy<0
(1) Both x and y are negative numbers=>xy>0=>The answer is NO=>SUF
(2) x=y => x and y have the same sign=>The answer is NO=>SUF
The answer is D

|xy|=-xy when xy<0, so the question asks whether xy<0 or not

(1) Both x and y are negative, hence xy>0 -> |xy| not equal to -xy -> Sufficient
(2) x-y=0 -> x=y -> xy>0 for all nonzero x, y -> Sufficient

Answer D

|xy| = -xy

Mod means the final value will always be positive. Hence we need to check scenarios for -XY

S1 Both x and y are negative numbers.,
- both are negative so " |xy| " will always be positive; in -xy , X will become positive, And Y is negative, so positive * negative = negative. Answer is definite "NO", no other scenario possible, sufficient.

S2 x - y = 0
It means both are equal and have and have opposite signs.
so |xy| will still remain postive because of mod;
but in -xy, there a two sceanrio, negative x & postive y or positive x & negative y
(1) negative x & postive y : -xy in this x becomes positive & y is positive as well, positive * positive = positive. Answer is definite "YES"
(2) positive x & negative y : -xy in this x becomes negative & y is negative as well, negative * negative = positive. Answer is definite "YES"
both scenario give same result, sufficient.

therfor Answer is D.

Asked: If x and y are nonzero numbers, does |xy| = -xy ?
|xy| = -xy if xy<0
Is xy < 0 ?

(1) Both x and y are negative numbers.
x<0; y<0; xy>0
xy is NOT <0
SUFFICIENT

(2) x - y = 0.
x = y
xy = x^2 > 0
xy is NOT <0
SUFFICIENT

IMO D

x, y = non-zero nos.
Q: |xy| = -xy?
|xy| = +ve => Q: Is xy = -ve?

1) x and y = -ve nos.
=> xy is positive ---> SUFFICIENT (xy is not negative)

2) x - y = 0
=> x = y => x and y have the same sign
=> xy is positive ---> SUFFICIENT (xy is not negative)

ANSWER = [D]

In order for -xy to be positive xy must be negative, thus one of them must be positive and the other negative.
(1) Same sign. Answer NO. Sufficient.
(2) It can be changed to x = Y. Same sign. Answer NO. Sufficient.
Each alone is sufficient.
D

IMO A

If x and y are nonzero numbers, does |xy| = -xy ?

(1) Both x and y are negative numbers.
(2) x - y = 0.

Statement 1

For both negative, LHS and RHS will both be opposite signs and hence will NOT be equal.
Hence SUFFICIENT

Statement 2:

x-y=0

All values of x & y holds true except 0. Hence not sufficient.

HENCE A