Is x + y < 1 ?

Is x + y < 1 ?
(1) x < 8/9
(2) y < 1/8 




























(1) Info only about x. Not sufficient
(2) Info only about y. Not sufficient
(1)+(2) x+y<73/72 Not sufficient

Answer: E.

Are some goats not cows

Are some goats not cows?
(1) All cows are lions
(2) All lions are goats.

 













This is good one:
Question generally asks is g>c?
(1) c<=l Not sufficient
(2) l<=g Not sufficient
(1)+(2) c<=l<=g --> If all cows are lions and all lions are goats there are no goat, which are not cows, in other case there are, so Not sufficient

Answer: E.

Is |x-1| < 1?
(1) (x-1)^2 <= 1
(2) x^2 - 1 > 0

Is |x-1| < 1? Basically the question asks is 0<x<2 true?

(1) (x-1)^2 <= 1 --> x^2-2x<=0 --> x(x-2)<=0 --> 0<=x<=2. x is in the range (0,2) inclusive. This is the trick here. x can be 0 or 2! Else it would be sufficient. So not sufficient.

(2) x^2 - 1 > 0 --> x<-1 or x>1. Not sufficient.

(1)+(2) Intersection of the ranges from 1 and 2 is 1<x<=2. Again 2 is included in the range, thus as x can be 2, we can not say for sure that 0<x<2 is true. Not sufficient.

Answer: E.

Is r=s?
(1) -s<=r<=s
(2) |r|>=s
 










-s<=r<=s, we can conclude two things from this statement:
A. s is either positive or zero, as -s<=s;
B. r is in the range (-s,s) inclusive, meaning that r can be -s as well as s.
But we don't know whether r=s or not. Not sufficient.

(2) |r|>=s, clearly insufficient.

(1)+(2) -s<=r<=s, s is not negative, |r|>=s --> r>=s or r<=-s. This doesn't imply that r=s, from this r can be -s as well.
Consider: s=5, r=5 --> -5<=5<=5 |5|>=5
s=5, r=-5 --> -5<=-5<=5 |-5|>=5
Both statements are true with these values. Hence insufficient.

Answer: E.

Is |x+y|>|x-y|?
(1) |x| > |y|
(2) |x-y| < |x|









To answer this question you should visualize it. We have comparison of two absolute values. Ask yourself when |x+y| is more then than |x-y|? If and only when x and y have the same sign absolute value of x+y will always be more than absolute value of x-y. As x+y when they have the same sign will contribute to each other and x-y will not.

5+3=8 and 5-3=2
OR -5-3=-8 and -5-(-3)=-2.

So if we could somehow conclude that x and y have the same sign or not we would be able to answer the question.

(1) |x| > |y|, this tell us nothing about the signs of x and y. Not sufficient.

(2) |x-y| < |x|, says that the distance between x and y is less than distance between x and origin. This can only happen when x and y have the same sign, when they are both positive or both negative, when they are at the same side from the origin. Sufficient. (Note that vise-versa is not right, meaning that x and y can have the same sign but |x| can be less than |x-y|, but if |x|>|x-y| the only possibility is x and y to have the same sign.)

Answer: B.

Is n'<'0

Is n<0?
(1) -n=|-n|
(2) n^2=16

St. (1) : -n = |-n|
Let -n = A ; therefore the statement becomes : A = |A|.
This can only be valid when A is positive (or equal to 0). This in turn means that n must be negative (or equal to 0).
-n=|-n| also for n=0, hence not sufficient.

St. (2) : n^2 = 16
n = ±4.
Thus Insufficient.



Answer : C

Is |a|/|b|=a/b

 a*b#0. Is |a|/|b|=a/b?
(1) |a*b|=a*b
(2) |a|/|b|=|a/b|

Question stem : Neither a nor b can hold the value 0 ; |a|/|b|=a/b
For condition to be true, both a and b must hold the same sign.

St. (1) : |a*b| = a*b
This condition will be satisfied only when both a and b are either both positive or both negative.
Hence Sufficient.

St. (2) : |a|/|b| = |a/b|
This condition can be satisfied when a and b are same sign as well as opposite sign.
Hence Insufficient.

Answer : A

|x+2|=|y+2| what is the value of x+y

 |x+2|=|y+2| what is the value of x+y?
(1) xy<0
(2) x>2 y<2

Question stem :
Note: Since the equations is symmetrical, there will only be two distinct cases. However, for the sake of explanation, I have illustrated all 4.
(a) When both x and y are greater than - 2 ; x + 2 = y + 2 ; x = y
(b) When both x and y are less than - 2 ; - x - 2 = - y - 2 ; x = y
(c) When x is less than -2 and y is greater than -2 ; - x - 2 = y + 2 ; x + y = - 4
(d) When x is greater than -2 and y is less than -2 ; x + 2 = - y - 2 ; x + y = - 4

St. (1) : xy < 0
This implies that one is negative and the other is positive. Therefore, in order for xy to be less than 0, x cannot be equal to y. Thus in order to satisfy the question stem, it can only be cases (c) and (d).
Thus Sufficient.

St. (2) : x > 2 ; y < 2
Again, this implies that x and y cannot be equal. Thus, in order to satisfy the question stem it can only be cases (c) and (d).
Thus Sufficient.

Answer : D