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In the xyplane shown, the shaded region consists of all points that l
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In the xyplane shown, the shaded region consists of all points that lie above the graph of y=x^2  4x and below the the xaxis. Does the point (a,b) (not shown) lie in the shaded region if b<0? (1) 0 < a < 4 (2) a^2  4a < b Source: Official GMAT Quantitative Review 2016 P. 162 DS #124 Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.) Attachment:
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Originally posted by dgboy765 on 23 Jan 2016, 13:47.
Last edited by Bunuel on 24 Jan 2016, 03:18, edited 1 time in total.
Edited the question and added the image.




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Re: In the xyplane shown, the shaded region consists of all points that l
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27 Jan 2016, 18:01
dgboy765 wrote: In the xyplane shown, the shaded region consists of all points that lie above the graph of y=x^2  4x and below the the xaxis. Does the point (a,b) (not shown) lie in the shaded region if b<0? (1) 0 < a < 4 (2) a^2  4a < b Source: Official GMAT Quantitative Review 2016 P. 162 DS #124 Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.) Dear dgboy765, I'm happy to help. Here are the basic ideas. Consider the graphs of the form y = [some expression of x]. These can include a) oblique lines, y = mx + b (e.g. y = (3/7)x + (5/7)) b) parabolae: e.g. y = x^2, or y = x^2  4c) higher powers of x: e.g. y = x^5  x^3d) square roots: y = sqrt(x  3) e) all kinds of other exotic curves: y = 2^x, or y = (1 + x)/(1  x), or etc. For the purpose of this discussion, the nature of that variety doesn't matter. What I am going to say applies to all graphs of the form y = [expression of x]. I will use the parabola y = x^2  4 as my example graph, but what I am saying about this graph applies to any graph of the form y = [expression of x] The first may be obvious: all the values (x, y) that satisfy the equation y = x^2  4 must lie exactly on the graph of y = x^2  4. More generally, the graph of any equation is the set of all ordered pairs that satisfy this equation. Now, think about any point on that line. Imagine we take an (x, y) that live on the graph, and we change the ycoordinate to make it bigger. That would result in a new point that is not on the graph but above the graph. This corresponds to the inequality y > x^2  4. Any ordered pair that satisfies this inequality is somewhere above the graph. Similarly, any ordered pair that satisfies the inequality y < x^2  4 lies somewhere below the graph. In this question, notice that for point (a, b), a is the xcoordinate and b is the ycoordinate. The shaded region shown in the picture is below the xaxis, so y < 0, and it is above the parabola y = x^2  4, so it is entirely defined in terms of these two inequalities: y < 0 and y > x^2  4 The prompt already specified that b < 0, and that takes care of the first inequality. Notice that statement #2 is just the second inequality written in terms of a & b instead of x & y. That's why it's sufficient. Once we specify both inequalities, we have specified the shaded region entirely. On a side note, notice that y = x, the line through the origin with a slope of 1, is the line that contains all points with equal x and ycoordinates. The inequality y > x are all the points above y = x, and the inequality x > y are all the points below y = x. Does all this make sense? Mike
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In the xyplane shown, the shaded region consists of all points that l
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27 Jan 2016, 21:24
dgboy765 wrote: In the xyplane shown, the shaded region consists of all points that lie above the graph of y=x^2  4x and below the the xaxis. Does the point (a,b) (not shown) lie in the shaded region if b<0? (1) 0 < a < 4 (2) a^2  4a < b Source: Official GMAT Quantitative Review 2016 P. 162 DS #124 Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.) Attachment: 20160124_1416.png Hi, In very simple terms to solve this Q..The type of parabola etc is amplified by mike in the above post... A parabola of Quadratic equation will have a minimum or a maximum value depending on the coeff of \(x^2\).. here it is positive, so the parabola will be open upwards and will have a mininmum value at\(x=\frac{b}{2a}\) or 4/2=2.. and the value is \(2^24*2=4\)... lets see the statements now..(1) \(0 < a < 4\).. nothing about b.... so point (a,b) can be anywhere depending on value of b at a=3.99, b can be 4, so will be outside the graph or at some point inside.. and at a=2, b can be 3.99, it will be inside the graph.. so insuff.. (2) \(a^2  4a < b\) the moment you see this equation, its similarity with the original equation y=x^2  4x should strike you.. we substitute a and b as x and y in the eq we get b=a^24a... since the equation \(b=a^24a\) is that of the the line.. a^24a< b will be inside the parabola and a^24a>b will be outside it... so suff.. you can test this with, say at the x axis.. at a=4, b=0.. \(a^24a=b... 4^24*4=b=0\).. so if \(a^24a<0, 0<a<4\) satisfies the condition for within the shaded portion and so suff.. the moment a^24a>0, a>5 or a<0 on x axis, and this point will be outside the parabola..hope it helped you in some way
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Re: In the xyplane shown, the shaded region consists of all points that l
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14 Sep 2016, 07:47
Hi Mike Actually, I can not get the concept behind your explanation. Please explain more taking into account the basic concepts since I did not master the Coordinate concepts. Thank you in advance.



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Re: In the xyplane shown, the shaded region consists of all points that l
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14 Sep 2016, 12:01
hatemnag wrote: Hi Mike Actually, I can not get the concept behind your explanation. Please explain more taking into account the basic concepts since I did not master the Coordinate concepts. Thank you in advance. Dear hatemnag, I'm happy to respond. My friend, before I answer your question, I am going to challenge you. I am going to challenge you to ask a better, more thorough question. See this blog article: Asking Excellent QuestionsYou see, the question you asked was very vague and general. Vague questions are poor questions. It's perfectly fine that you don't understand some basic concepts about Coordinate Geometry and that you want to learn morein fact, it's wonderful that you are asking for help! The trouble is, I have absolutely no idea what you already know and what you need. Your vague question completely leaves me in the dark. An excellent question would involve making explicitly clear, exactly and specifically, what parts you understand and what parts you don't understand. For example, you could go through each point I make in my Jan. 27, 2016 post above, and tell me exactly what you understand and exactly what confuses you about each item. Of course, a more specific, more detailed question, will help me respond to you more effectively, but what students often fail to understand: I recommend crafting an excellent question, not for me, but for you! The process of putting all the effort into writing an excellent question will be tremendously helpful to you: by explaining all this, you will force yourself to make connections and have realizations, and all this effort will prime your mind so that my more detailed response will be that much more helpful for you. It's much harder to produce an excellent question, and, in fact, all the effort it takes to craft an excellent question is actually an investment in your own understanding and learning. When a student poses an excellent question to the teacher, it's a total winwin scenario. This is precisely why asking excellent questions is one of the habits of excellence. Therefore, my friend, I am going to challenge you to write an excellent question. Craft the highest quality question you can, making explicitly and specifically clear precisely which parts you understand and don't understand about coordinate geometry. It's fine to have a ton of questions, that's great, but it's important for both of us to appreciate the basic ideas that you do understand, because all learning is based on what you already grasp. Does all this make sense? Mike
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Re: In the xyplane shown, the shaded region consists of all points that l
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12 Feb 2017, 22:13
hi there, mikemcgarry and chetan2u can someone help me out? I am taking the GMAT in 8 days (my practice CATs have been in the 700 range Q49, V40), but this question came as a bit of a shock to me. Note, I have not read either of your explanations yet because this concept looks new to me. What is "a" and "b" referring to? Is it the a and b within the quadratic equation (ax^2+bx+c=0)? For the most part, the only questions I have seen related to graphs in my practice is lines (y=mx+b). I will read your explanations shortly, but first, is there a gap in my understanding of the basics? I think I remember from grade school this U shape being a "Parabola." Is that correct? If so, what about Parabola's do we need to know for the GMAT?



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Re: In the xyplane shown, the shaded region consists of all points that l
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13 Feb 2017, 05:08
gzimmer wrote: hi there, mikemcgarry and chetan2u can someone help me out? I am taking the GMAT in 8 days (my practice CATs have been in the 700 range Q49, V40), but this question came as a bit of a shock to me. Note, I have not read either of your explanations yet because this concept looks new to me. What is "a" and "b" referring to? Is it the a and b within the quadratic equation (ax^2+bx+c=0)? For the most part, the only questions I have seen related to graphs in my practice is lines (y=mx+b). I will read your explanations shortly, but first, is there a gap in my understanding of the basics? I think I remember from grade school this U shape being a "Parabola." Is that correct? If so, what about Parabola's do we need to know for the GMAT? Hi gzimmer, a and b refer to a point in the plane where a is on xaxis and B on yaxis.. You are correct that such equation refers to parabola. Also parabola if asked would be basic, as for instance one can see the similarities in two equations one in Q and other in statement II. And that would also mean you are likely to be doing well in Quant till then. Now with just 8 days left, just do the basic and do not spend too much time on topics rarely seen .. All the best.
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Re: In the xyplane shown, the shaded region consists of all points that l
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13 Feb 2017, 16:16
dgboy765 wrote: In the xyplane shown, the shaded region consists of all points that lie above the graph of y=x^2  4x and below the the xaxis. Does the point (a,b) (not shown) lie in the shaded region if b<0? (1) 0 < a < 4 (2) a^2  4a < b Source: Official GMAT Quantitative Review 2016 P. 162 DS #124 Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.) Attachment: 20160124_1416.png Hi, I solved it this way: remember that stem says b<0. st1. clearly insufficient because say y= 5 then it is not in shade region but for y greater that 4 it is in region. st2. you can rewrite it as a^24a+4<b+4 (add 4 to both sides) then LHS is (a2)^2 and RHS is b+4 > (a2)^2<a+4 as LHS is something always positive then it can not be smaller than something negative thus b+4 must be greater than 0, therefore we have 4<b<0 hence 0<b+4<4. > (a2)^2<4 > x2<2 > 2<x2<2 > 0<x<4. it is in the region. suff. B



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Re: In the xyplane shown, the shaded region consists of all points that l
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06 Mar 2017, 11:39
i am not able to understand the solution can anyone please help



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Re: In the xyplane shown, the shaded region consists of all points that l
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06 Mar 2017, 11:47
Psk13 wrote: i am not able to understand the solution can anyone please help My friend, there are very good solutions above. Please elaborate what did not you understand. Thank you.
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Re: In the xyplane shown, the shaded region consists of all points that l
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05 Jun 2017, 07:43
chetan2u wrote: dgboy765 wrote: In the xyplane shown, the shaded region consists of all points that lie above the graph of y=x^2  4x and below the the xaxis. Does the point (a,b) (not shown) lie in the shaded region if b<0? (1) 0 < a < 4 (2) a^2  4a < b Source: Official GMAT Quantitative Review 2016 P. 162 DS #124 Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.) Attachment: 20160124_1416.png Hi, In very simple terms to solve this Q.. The type of parabola etc is amplified by mike in the above post... A parabola of Quadratic equation will have a minimum or a maximum value depending on the coeff of x^2.. here it is positive, so the parabola will be open upwards and will have a mininmum value at x=b/2a or 4/2=2.. and teh value is 2^24*2=4... lets see the statements now.. (1) 0 < a < 4.. nothing about b.. at a=3.99, b can be 4, so will be outside the graph .. and at a=2, b can be 3.99, it will be inside the graph.. so insuff.. (2) a^2  4a < b the moment you see this equation, its similarity with the original equation y=x^2  4x should strike you.. we substitute a and b as x and y in the eq we get b=a^24a... since the equation b=a^24a is that of the the line.. a^24a< b will be inside the parabola and a^24a>b will be outside it... so suff.. you can test this with, say at the x axis.. at a=4, b=0.. a^24a=b... 4^24*4=b=0.. so if a^24a<0, 0<a<4 satisfies the conditi0n for within the shaded portion and so suff.. the moment a^24a>0, a>5 or a<0 on x axis, and this point will be outside the parabola.. hope it helped you in some way In Statement 1 couldn't Point B be anything? For instance 100000 ? little bit confused why you picked 3.99 and and 4 for b ? or am I missing something ? Im sure Im missing something or the questions is just very basic I mean from statement 1: no information about b at all > so clearly insufficient. Statement 2: it explicitly tells us that a,b lies above the graph and it is given that b<0
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Re: In the xyplane shown, the shaded region consists of all points that l
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18 Aug 2017, 18:56
Hi VeritasPrepKarishma mikemcgarry shashankism BunuelCan you suggest flaw in below approach: Given: Question has asked me whether a point (a,b) lies on shaded curve. I hope there is no discrepancy between whether the shaded curve includes the parabola. As per mine understanding it does not. Essentially the parabola is in quadrant IV and hence (x,y) should be coordinates of points in this quadrant. St 1: 0<a<4 > I simply took in random values between 0 to 3.9 and found that since x is positive st 1 is sufficient. After reading the solutions and above explanation found that for two different values of a/x, say 1 or 8 I am not alway getting positive values of y hence st 1 is insufficient. St 2: at first glance the equation translates to x^2  4x < y since (a,b) have to in shaded region but for these how did we intuitively assume that (a,b) is on curve? this query is especially pertaining to above explanation by mikemcgarryAs per my understanding it seems we have put an upper gap on parabola by saying b<0 in question stem. Any view on this?
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Re: In the xyplane shown, the shaded region consists of all points that l
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20 Aug 2017, 02:19
daviddaviddavid wrote: chetan2u wrote: dgboy765 wrote: In the xyplane shown, the shaded region consists of all points that lie above the graph of y=x^2  4x and below the the xaxis. Does the point (a,b) (not shown) lie in the shaded region if b<0? (1) 0 < a < 4 (2) a^2  4a < b Source: Official GMAT Quantitative Review 2016 P. 162 DS #124 Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.) Attachment: 20160124_1416.png Hi, In very simple terms to solve this Q.. The type of parabola etc is amplified by mike in the above post... A parabola of Quadratic equation will have a minimum or a maximum value depending on the coeff of x^2.. here it is positive, so the parabola will be open upwards and will have a mininmum value at x=b/2a or 4/2=2.. and teh value is 2^24*2=4... lets see the statements now.. (1) 0 < a < 4.. nothing about b.. at a=3.99, b can be 4, so will be outside the graph .. and at a=2, b can be 3.99, it will be inside the graph.. so insuff.. (2) a^2  4a < b the moment you see this equation, its similarity with the original equation y=x^2  4x should strike you.. we substitute a and b as x and y in the eq we get b=a^24a... since the equation b=a^24a is that of the the line.. a^24a< b will be inside the parabola and a^24a>b will be outside it... so suff.. you can test this with, say at the x axis.. at a=4, b=0.. a^24a=b... 4^24*4=b=0.. so if a^24a<0, 0<a<4 satisfies the conditi0n for within the shaded portion and so suff.. the moment a^24a>0, a>5 or a<0 on x axis, and this point will be outside the parabola.. hope it helped you in some way In Statement 1 couldn't Point B be anything? For instance 100000 ? little bit confused why you picked 3.99 and and 4 for b ? or am I missing something ? Im sure Im missing something or the questions is just very basic I mean from statement 1: no information about b at all > so clearly insufficient. Statement 2: it explicitly tells us that a,b lies above the graph and it is given that b<0 hi.. since b is not given and no relation exists between a and b, b can take any value.. you are correct in your understanding...two points given were just to illustrate that b could be inside or outside graph
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Re: In the xyplane shown, the shaded region consists of all points that l
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26 Apr 2018, 10:07
I actually spent a lot of time on this simple question. Here is the mistake I did for anyone who is still confused: I assumed that "a" is on x22x. Therefore, when I saw condition (1), 0<a<4, I started plugging in values for a in X22x . This would have worked if the assumption was stated; however, at no point did the question say that a is on function x22x. Therefore, we can not determine "b" from "a" because we are not given any info that ties the two point together.
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In the xyplane shown, the shaded region consists of all points that l
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26 Apr 2018, 23:38
For any x value there are corresponding 2 y values, so Statement I cannot be sufficient..... Statement 2 . y < x^2  4x, this actually means the region outside the parabola.. Sufficient.
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Re: In the xyplane shown, the shaded region consists of all points that l
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07 May 2018, 11:46
chetan2u wrote: dgboy765 wrote: In the xyplane shown, the shaded region consists of all points that lie above the graph of y=x^2  4x and below the the xaxis. Does the point (a,b) (not shown) lie in the shaded region if b<0? (1) 0 < a < 4 (2) a^2  4a < b Source: Official GMAT Quantitative Review 2016 P. 162 DS #124 Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.) Attachment: 20160124_1416.png Hi, In very simple terms to solve this Q..The type of parabola etc is amplified by mike in the above post... A parabola of Quadratic equation will have a minimum or a maximum value depending on the coeff of \(x^2\).. here it is positive, so the parabola will be open upwards and will have a mininmum value at\(x=\frac{b}{2a}\) or 4/2=2.. and the value is \(2^24*2=4\)... lets see the statements now..(1) \(0 < a < 4\).. nothing about b.... so point (a,b) can be anywhere depending on value of b at a=3.99, b can be 4, so will be outside the graph or at some point inside.. and at a=2, b can be 3.99, it will be inside the graph.. so insuff.. (2) \(a^2  4a < b\) the moment you see this equation, its similarity with the original equation y=x^2  4x should strike you.. we substitute a and b as x and y in the eq we get b=a^24a... since the equation \(b=a^24a\) is that of the the line.. a^24a< b will be inside the parabola and a^24a>b will be outside it... so suff.. you can test this with, say at the x axis.. at a=4, b=0.. \(a^24a=b... 4^24*4=b=0\).. so if \(a^24a<0, 0<a<4\) satisfies the condition for within the shaded portion and so suff.. the moment a^24a>0, a>5 or a<0 on x axis, and this point will be outside the parabola..hope it helped you in some way Hi, thanks for your explanation. I have a doubt In condition B  I understand that a^24a< b will be inside the parabola and a^24a>b will be outside it. However, the question asks us about the shaded region. Whereas a^24a< b will include all area inside the parabola  both above x axis and below x axis. this to ensure that the point lies under the x axis we will need Condition A (Which we can't prove/ derive from Condition B) Therefore I feel Answer should be C. Please let me know if I am missing out on something



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Re: In the xyplane shown, the shaded region consists of all points that l
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07 May 2018, 22:46
Tanvi94 wrote: chetan2u wrote: dgboy765 wrote: In the xyplane shown, the shaded region consists of all points that lie above the graph of y=x^2  4x and below the the xaxis. Does the point (a,b) (not shown) lie in the shaded region if b<0? (1) 0 < a < 4 (2) a^2  4a < b Source: Official GMAT Quantitative Review 2016 P. 162 DS #124 Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.) Attachment: 20160124_1416.png Hi, In very simple terms to solve this Q..The type of parabola etc is amplified by mike in the above post... A parabola of Quadratic equation will have a minimum or a maximum value depending on the coeff of \(x^2\).. here it is positive, so the parabola will be open upwards and will have a mininmum value at\(x=\frac{b}{2a}\) or 4/2=2.. and the value is \(2^24*2=4\)... lets see the statements now..(1) \(0 < a < 4\).. nothing about b.... so point (a,b) can be anywhere depending on value of b at a=3.99, b can be 4, so will be outside the graph or at some point inside.. and at a=2, b can be 3.99, it will be inside the graph.. so insuff.. (2) \(a^2  4a < b\) the moment you see this equation, its similarity with the original equation y=x^2  4x should strike you.. we substitute a and b as x and y in the eq we get b=a^24a... since the equation \(b=a^24a\) is that of the the line.. a^24a< b will be inside the parabola and a^24a>b will be outside it... so suff.. you can test this with, say at the x axis.. at a=4, b=0.. \(a^24a=b... 4^24*4=b=0\).. so if \(a^24a<0, 0<a<4\) satisfies the condition for within the shaded portion and so suff.. the moment a^24a>0, a>5 or a<0 on x axis, and this point will be outside the parabola..hope it helped you in some way Hi, thanks for your explanation. I have a doubt In condition B  I understand that a^24a< b will be inside the parabola and a^24a>b will be outside it. However, the question asks us about the shaded region. Whereas a^24a< b will include all area inside the parabola  both above x axis and below x axis. this to ensure that the point lies under the x axis we will need Condition A (Which we can't prove/ derive from Condition B) Therefore I feel Answer should be C. Please let me know if I am missing out on something Hello The point you make is valid, that a^2  4a < b will include all area inside the parabola, both above x axis and below x axis. But its also mentioned in the question stem that b < 0. Since y coordinate is negative, the area that we have to consider is the one below xaxis only.



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Re: In the xyplane shown, the shaded region consists of all points that l
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29 Jun 2018, 10:19
mikemcgarry wrote: dgboy765 wrote: In the xyplane shown, the shaded region consists of all points that lie above the graph of y=x^2  4x and below the the xaxis. Does the point (a,b) (not shown) lie in the shaded region if b<0? (1) 0 < a < 4 (2) a^2  4a < b Source: Official GMAT Quantitative Review 2016 P. 162 DS #124 Can someone explain the process to solving this problem in the simplest way possible? (but please don't be overly brief. I'm not as intuitive as you.) Dear dgboy765, I'm happy to help. Here are the basic ideas. Consider the graphs of the form y = [some expression of x]. These can include a) oblique lines, y = mx + b (e.g. y = (3/7)x + (5/7)) b) parabolae: e.g. y = x^2, or y = x^2  4c) higher powers of x: e.g. y = x^5  x^3d) square roots: y = sqrt(x  3) e) all kinds of other exotic curves: y = 2^x, or y = (1 + x)/(1  x), or etc. For the purpose of this discussion, the nature of that variety doesn't matter. What I am going to say applies to all graphs of the form y = [expression of x]. I will use the parabola y = x^2  4 as my example graph, but what I am saying about this graph applies to any graph of the form y = [expression of x] The first may be obvious: all the values (x, y) that satisfy the equation y = x^2  4 must lie exactly on the graph of y = x^2  4. More generally, the graph of any equation is the set of all ordered pairs that satisfy this equation. Now, think about any point on that line. Imagine we take an (x, y) that live on the graph, and we change the ycoordinate to make it bigger. That would result in a new point that is not on the graph but above the graph. This corresponds to the inequality y > x^2  4. Any ordered pair that satisfies this inequality is somewhere above the graph. Similarly, any ordered pair that satisfies the inequality y < x^2  4 lies somewhere below the graph. In this question, notice that for point (a, b), a is the xcoordinate and b is the ycoordinate. The shaded region shown in the picture is below the xaxis, so y < 0, and it is above the parabola y = x^2  4, so it is entirely defined in terms of these two inequalities: y < 0 and y > x^2  4 The prompt already specified that b < 0, and that takes care of the first inequality. Notice that statement #2 is just the second inequality written in terms of a & b instead of x & y. That's why it's sufficient. Once we specify both inequalities, we have specified the shaded region entirely. On a side note, notice that y = x, the line through the origin with a slope of 1, is the line that contains all points with equal x and ycoordinates. The inequality y > x are all the points above y = x, and the inequality x > y are all the points below y = x. Does all this make sense? Mike Hi ! Just want to reconfirm : If it was in inverted parabola ... ie lets say with an equation y=(x^2)4 , in that case the points lying inside the parabola and the points lying outside the parabola would be defined as : Points lying inside the parabola y > (x^2)4 Points lying outside the parabola y < (x^2)4



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Re: In the xyplane shown, the shaded region consists of all points that l
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26 Oct 2018, 08:53
(1) 0<a<4. b can be any value > insufficient.
(2) a^2  4a < b Because b<0 > a^2  4a<0 > 0<a<4 (*)
If a point (a,b') lies on the paragraph y=x^2  4x, it satisfies the equation a^2  4a = b' > b'<b<0. It means that b is between b' and 0 (**) From (*) and (**) > The point (a,b) lies in the shaded region.



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Re: In the xyplane shown, the shaded region consists of all points that l
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16 Dec 2018, 07:46
Hi Guys, I have spent too many minutes on this questions which is also on OG quant review 2019 DS01613, I am still not able to get through the questions ask and the explanation given here. I think i have a thick brain it seems, Can Bunuel or VeritasKarishma help me in understanding the question stem and theory/concept behind it. Thanks




Re: In the xyplane shown, the shaded region consists of all points that l
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16 Dec 2018, 07:46



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