Every point in the xy plane satisfying the condition ax + by ≥ c is sa

rockroars: I appreciate the effort for the diagrams. But you made a tiny judgment error. Let me explain the answer in detail.

First of all, notice that ax + by – c = 0 or ax + by = c is the equation of the same line. A line divides the plane into two regions. One of them, where every point (x, y) satisfies ax + by ≥ c, is region R.

Statement 1: Slope of line is 2

the line will pass through third quadrant and hence both regions will lie in the third quadrant. Sufficient.

Statement 2: The line passes through (-3, 0).

A line passing through (-3, 0) could be the blue line or the green line. In either case, the line will pass through the third quadrant and hence, will have both regions in the third quadrant. So it is sufficient too? What about the x axis? That is also a line passing through (-3, 0). It does not pass through the third quadrant. We would need the equation of the line to find out whether our region R lies in the third quadrant. The equation of x axis is y = 0. So the required region is y ≥ 0 i.e. the first and second quadrant. Hence using just this information, we cannot say whether a point of region R lies in the third quadrant or not.
Answer (A)

Any line with a positive slope will pass from 1st and 3rd quadrant, and either "one of the other two quadrants" or the "origin".

S1: Slope is positive. Sufficient.
S2: Line passes from (-3,0) so the line either passes from 3rd Quadrant or the equation is of X-axis. Not Sufficient.

Answer: A

A line (which by definition, extends infinitely at both sides) given by ax+by - c = 0 splits a region into two sections - one on the left side of the line and the other on the right side of the side. One of these two regions will satisfy ax+by - c < 0 and the other will satisfy ax+by - c > 0.
How do you know which region satisfies which inequality? Put a point from that region in the inequalities and see what it satisfies e.g. if (0, 0) doesn't lie on the line, put x = 0, y = 0
If c is negative, ax+by - c < 0 will be satisfied and the region that contains the point (0, 0) i.e. the origin of the axis will satisfy ax+by - c < 0.
In that case, the other region will satisfy ax+by - c > 0.

Which quadrants will lie in any particular region depends on where the line is located. If it is a vertical line passing through first and fourth quadrant, second and third quadrant will lie to its left and some part of first and fourth quadrants will also lie to its left. Rest of the first and fourth quadrants will lie to its right (make a line and see what i mean)
Similarly, try making a line with a positive slope, say 2. It will pass through first and third quadrants in ALL cases (remember, a line extends indefinitely at both ends). Since it will pass through the third quadrant, both the regions will have a part of the third quadrant.

Is it A ?
I think A is sufficient becase if the slope is 2, then some solutions lie in the III quadrant.
B seems to me insufficient because I need at least the coordinates of 1 more point to find the slope.

A : Sufficient ... Since slope is 2, hence positive. This means that the line must pass through the third quadrant. The inequality ax+by>=c represents one side of the line. Since the line passes through the 3rd quadrant, either side has points from the 3rd quadrant. Hence the region will always havea bit of third quadrant.

B : Insuffcient ... Any line except the one with slope=0 will pass through 3rd quadrant and the above logic applies. But the line with slope 0 may or may not have the thrid quadrant points in the region included depending on which side we choose (sign of c)

Answer : A

I think the answer should be C.


We get the details of line and regions completely only after we club 1 and 2. After which we will be in a position to decide if Region R passes through 3rd Quad.

I am used to perfect answers from you shrouded1... But I think you missed out on a point here. c = 0 we know because it has to be x axis since it passes through (-3, 0). Since y >= 0 is the first and second quadrant hence we know that the region R may or not lie in third quadrant. If instead, we had ax+by <=c, statement 2 would be sufficient too.
Nonetheless, your answer is correct.

If I am missing something here, let me know. (I would like to believe that I didn't err in a question I made myself!)

@VeritasPrepKarishma:

Okay! I realized where I made a mistake, a line with slope 2 is acute to x-axis. I assumed it to be obtuse to X axis.

Thanks a lot for your reply

Karishma: I didn't understand. Could you explain a bit.

Let's try this relatively simple question:

Every point in the xy plane satisfying the condition ax + by ≥ c is said to be in region R. If a, b and c are real numbers, does any point of region R lie in the third quadrant?
1. Slope of the line represented by ax + by – c = 0 is 2.
2. The line represented by ax + by – c = 0 passes through (-3, 0).

This is bit tricky question.
You have to know that a line with positive slope makes an acute angle with the x axis or it always passes through 1 and 3rd quadrant. This is like a theorm. Helpful in some cases. Since Slope is +. It gives the soln.

2.Now here you have look back at the question and see the value of a,b and c. The points a,b and c will always pass through 3rd quadrant barring one case. In case the line is x - axis itself. ie x = 0 . and c =0 and b = 1.

Hence A gives a clear soln.

https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2010/12 ... s-part-ii/ is helpful.

Thank you.

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