If y is an integer, is y

Question

If y is an integer, is  y\leq{-1} ?

(1)  x+z^2\leq{0} 
(2)  y<x

TestPrepUnlimited · Answer

Statement 1:

We may write this as  y^2\leq{-x}  so  -x  must be positive or 0, hence  x  must be negative or 0. Insufficient.

Statement 2:

Insufficient.

Combined:

We did note x must be negative or 0 earlier, and y must be less than x. Then we should have  y \leq{-1} . (Sufficient)

However if we dig deeper, add  y^2  to both sides of (2), then we have  y + y^2 < x + y^2 \leq{0} . Thus  y*(y + 1) < 0 . We cannot have y = 0 or y = -1 as that makes the inequality 0 < 0. Any other positive y or negative y values make the left side bigger than 0. Thus there is actually no solution for y (and the question needs a small fix).

chetan2u · Answer

You are absolutely correct. 
Statement I requires to be changed.

PgRaul · Answer

If y is an integer, is y≤−1?

(1) x+z2≤0
(2) y<x

1. z^2 always a positive value. even if z= 0 or -ve or +ve, So if x + z^2 <= 0, x= -ve or 0, Insufficient
2. y/,x, insufficient
on combining y will be always negative. So if x = 0, y = -1, if x = -1, y = -2. As y takes only integer values. In any case y is always <= -1. 
Ans : C

palaknayyar · Answer

Statement 1: No mention of y at all and there are no specific hints from the question stem itself either. No sufficient.

Statement 2: Since we do not know anything about x, y can be any integer. Not sufficient.

Combining the 2: From S1, z^2 has to be positive, so for x+z^2 to be <0, x needs to be a negative number with a greater magnitude than z^2. From S2, clearly, y<x so y has to be equal to or < -1

Final answer : C

Rtejpar · Answer

palaknayyar , can we assume that z is an integer?

if z=.1, then y <=-.01, which does not help/would make the answer E.

However, if we assume z is an integer, then we can figure that z^2 >= 1, which then, of course, y=<-1 and we can answer C...

thoughts?

chetan2u · Answer

If z=01, then  x\leq -0.01<0

So y<x<0.
Since y is an integer, y is at the most -1 =>  y\leq -1

If y is an integer, is y <= -1?

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