1st statement is enough to ans the question. The 1st statement alone gives ans D as x can be less than -6 or more than 6 however since we are also given with 2nd statement x can be only less than -6.
We have to take into account all the informtion given prior to the quantities we are comparing.

Posted from my mobile device

Since x*y^2 < 0 and y is an integer,

Since y^2 > 0,
x < 0

And based on |-x| >= 6, we can see that x <-6, therefore B is the answer.

\(|-x| \geq 6\)

\(x*y^2 < 0\) and \(y\) is an integer.

Let us look at the second inequality

\(x*y^2 < 0\) and \(y\) is an integer

Clearly, this states that \(x\) has to be negative, otherwise this inequality will not be satisfied.

Let us look at the first inequality

\(|-x| \geq 6\)

This clearly states that the magnitude of \(x\) has to be \(6\).

Thus the first inequality fixes the magnitude of \(x\)
and the second inequality fixes its sign.

Therefore, \(x = -6\)

Clearly, Quantity B is greater.

This was given so that we understand to restrict the possible values of x to the negative.
The first statement given gives you two ranges:
x>=6, and x<=-6. By telling you that x*y^2<0, you know that y^2 is some positive integer, so for that number to be negative x must be negative.

Why is is true that the magnitude must be equal to 6?