No shortcuts. Pick numbers for x and see the boundaries.

Similar to this

https://gre.myprepclub.com/forum/in-the-xy ... tml#p30106

Or this explanation https://gre.myprepclub.com/forum/in-the-xy ... tml#p30856

Consider drawing this. You will have a triangular region bounded between -6 and 6, and above the X axis.
Since y must be positive, you can eliminate immediately choices E and F are gone and since the bounds are between -6 and 6, choice A is gone as well.
Choice B is certainly possible, since this would just be on the x axis
Choice C is possible, since for x=2, y=4, so (2,3) is inside the triangle
Choice D is not possible, since for x=3, y=3, so 3,4 would be just above the triangle
Choice G is possible, since for x=-3 the bound is y=3, and -3,1 is above the x-axis and under the triangle.

Answer should be B,C and G - Carcass I am not seeing answer choices here so I am not sure.

I believe is correct what you say sir

the x axis is not mentioned and could be also positive BUT the y axis must be positive and for being so our x in y-6|x| must be between 0<x<6 included

y=6-0=6 upper value

and

y=6-6=0 lowest value

So we have to include only the answer choices that has those values for y and the x both positive and negative

Therefore your answers are correct

Here a grate explanation for a similar question

https://gre.myprepclub.com/forum/in-the ... tml#p30856

Made the graph in Desmos, as it easily shows what I described: