Here,
the line with co-ordinate passes through point (0,0) and (-4,4)

Hence the slope of the line = -1

Now we can see the point (p,q) lie below the slope of line of slope -1 and will have slope < -1

However, the point (a,b) lies above the line with the slope of -1 and it will have a slope > -1

Therefore,

QTY A > QTY B

Experts plz comment.

How can we assume that the line runs through the origin?

Yes I agree, I not only think it must be stated it goes through the origin but in this case, to my eyes it actually seems to be a little off of it

The line passes through (-4,4) and (0,0) slope comes about as -1
Hence, the line is forming 45 degree between x and y plane for any quadrant.
Now, the crucial information is every point on the line will have the same numeric x and y coordinates.
In the second quadrant x will bear -ve value while in the 4th quadrant y will be -ve value therefore,

For (P,Q) P is slightly to the left of the line hence it is more -ve when compared to the parallel position within the line while in the (a,b) scenario a is slightly to the right of the line and hence more +ve than had the point been parallely placed in the line itself.

Therefore, p+q will always be -ve while a+b will always be +ve.

Two questions:

1.


Is the fact that y>-x for TOP region and y<-x for BOTTOM region only true for line equations y = -x or is it true for ALL lines in the XY plane?

2.


If we didn't have the line with equation y = -x (and had some other equation say y = 3x-2 or y = -3x - 2), would the above boldfaced portion still be true?

For every line in a 2D plane ax + by + c = 0. All the points in the 2D plane can be covered using the following conditions:
ax + by +c > 0 for all points above the line
ax + by + c = 0 for all the points on the line
ax + by +c < 0 for all points below the line

So, y>-x for TOP region and y<-x for BOTTOM region is true only for the line y = -x, but for other lines you can use the concept explained above.


2.




If we did not have y = -x line and had let's say y = 3x-2 ( or y - 3x + 2 = 0) then the conditions will change as follows
y - 3x + 2 > 0 for all points above the line
y - 3x + 2 = 0 for all the points on the line
y - 3x + 2 < 0 for all points below the line

And if you substitute the point (p, q) in it you will get
q - 3p + 2 < 0 (As the point is below the line)

And for (a, b) you will get
b - 3a + 2 > 0 as the point is above the line.

Hope it helps!

GreenlightTestPrep how do we decide which side is bottom half and which side is top half?

a +ve
b -ve
a>b so a+b will be positive
p -ve
q +ve
p>q so p+q will be negative
so option A