Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 60627

In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
03 Mar 2014, 01:21
Question Stats:
62% (01:46) correct 38% (01:58) wrong based on 1096 sessions
HideShow timer Statistics
In the rectangular coordinate system shown above, which quadrant, if any, contains no point (x,y) that satisfies the inequality \(2x  3y\leq{ 6}\) ? (A) None (B) I (C) II (D) III (E) IV Problem Solving Question: 123 Category: Geometry Simple coordinate geometry Page: 77 Difficulty: 600 The Official Guide For GMAT® Quantitative Review, 2ND EditionAttachment:
Untitled.png [ 7.91 KiB  Viewed 23494 times ]
Official Answer and Stats are available only to registered users. Register/ Login.
_________________




Math Expert
Joined: 02 Sep 2009
Posts: 60627

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
03 Mar 2014, 01:21
SOLUTIONIn the rectangular coordinate system shown above, which quadrant, if any, contains no point (x,y) that satisfies the inequality \(2x  3y\leq{ 6}\) ?(A) None (B) I (C) II (D) III (E) IV \({2x3y}\leq{6}\) > \(y\geq{\frac{2}{3}x+2}\). Thi inequality represents ALL points, the area, above the line \(y={\frac{2}{3}x+2}\). If you draw this line you'll see that the mentioned area is "above" IV quadrant, does not contains any point of this quadrant. Else you can notice that if \(x\) is positive, \(y\) can not be negative to satisfy the inequality \(y\geq{\frac{2}{3}x+2}\), so you can not have positive \(x\), negative \(y\). But IV quadrant consists of such \((x,y)\) points for which \(x\) is positive and \(y\) negative. Thus answer must be E. Answer: E.
_________________




Intern
Joined: 22 Jun 2013
Posts: 34

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
04 Mar 2014, 03:07
My answer is Option E i.e. 4th Quadrant To draw the line on the Coordinate system consider the Inequality : \(2x  3y\leq{ 6}\) as \(2x  3y={ 6}\) We get points (0,2) & (3,0 ) So the line looks some what as in the attachment. To find out which area is covered by the graph put the Cordinate (0,0) in the original question \(2x  3y\leq{ 6}\) We get: \(0 \leq {6}\). Which is False. So (0,0) does not lie in the area covered by the graph, Therefore the equation covers the area above the line. Thus 4th Quadrant does not contain any point that satisfies the inequality. ( Rest 3 Quadrants will have a few points that would satisfy the inequality) Not experienced enough to comment on the difficulty level.
Attachments
Q.png [ 21.53 KiB  Viewed 18334 times ]




Director
Joined: 25 Apr 2012
Posts: 648
Location: India
GPA: 3.21
WE: Business Development (Other)

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
09 Mar 2014, 03:28
Attachment: The attachment Untitled.png is no longer available In the rectangular coordinate system shown above, which quadrant, if any, contains no point (x,y) that satisfies the inequality \(2x  3y\leq{ 6}\) ?
(A) None (B) I (C) II (D) III (E) IVSol: The inequality can be 2x=3y=6 can be rewritten in Y intercept form as y=2x/3 +2 Attachment:
Untitled1.png [ 11.93 KiB  Viewed 18151 times ]
Notice that the slop of the line is positive and hence it will definitely pass through Quad 1 and 3. If the slope was negative then the line will definitely pass through Quad 2 and 4. Now for x=0, y= 2 that means line will have to pass through Quad 2 as well. Hence No point in Quad 4 will satisfy the given equation.
_________________
“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”



Intern
Joined: 09 Feb 2013
Posts: 22

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
18 Apr 2015, 11:53
Guys, I am still not clear with the solution. By any way can you simplify it further ?



eGMAT Representative
Joined: 04 Jan 2015
Posts: 3219

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
21 Apr 2015, 04:28
kshitij89 wrote: Guys, I am still not clear with the solution. By any way can you simplify it further ? Hi kshitij89  For every point lying on the line segment \(2x  3y =6\), the x and y coordinates are such that subtracting 3 times the y coordinate from 2 times the x coordinate is equal to 6. Examples are (3,4), (6,6) etc. For any other point not lying on this line segment, this difference of 2x and 3y is either less than 6 or greater than 6. The question asks us to find the location of points for which \(2x  3y <= 6\) is not true i.e. points for which the difference of 2x and 3y is not less than or equal to 6. For finding such points, we need to first plot its equivalent line segment in the XY coordinate system. Let's see how we can plot the line. We need two points for plotting the line segment.We know that for all points on the Xaxis, their Y coordinate is 0 and vice versa. So, putting x =0 in the equation of the line segment, we get the value of y = 2 and for y =0, we get the value of x = 3. Now, we have the x and the y  intercept for the line segment. Using this information, we can plot the line segment as shown below: Now, we will need some test case to establish that on which side of the line the points do not satisfy the inequality \(2x  3y <= 6\). The best way is to test for the intersection point of X & Y axis i.e. (0, 0). If we put x =0 and y =0 in the inequality, we get \(0 <=6\) which is not true. So, we can say with certainty that the side of the line which contains ( 0, 0) does not satisfy the inequality. This would mean that the area on the left side of the line segment \(2x  3y =6\) satisfy the inequality \(2x  3y <= 6\) and the area on the right side of the line segment do not satisfy the inequality. Since the question talks about such regions in terms of quadrants, we can observe that area on the left side of the line segment passes through Q I, II & III only. So, we can say for sure that Q IV does not contain any point that satisfy the inequality. Hope this is clear. Let me know if you still have trouble in understanding of the solution. Regards Harsh
_________________



Intern
Joined: 09 Feb 2013
Posts: 22

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
21 Apr 2015, 10:52
Hi Harsh,
Thanks for your reply.
The explanation was crisp and clear but I am not sure if I will be able to solve similar questions with different nos.
Can you please provide examples of similar questions to further test my understanding ?
Regards Kshitij



EMPOWERgmat Instructor
Status: GMAT Assassin/CoFounder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15965
Location: United States (CA)

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
21 Apr 2015, 19:59
Hi kshitij89, How are your overall graphing "skills"? Are you comfortable with the basic concepts, formulas, drawing graphs, etc.? If so, then you'll probably handle the concept on Test Day just fine. "Graphing", as a category, is relatively rare on the GMAT  you'll likely see just 12 graphing questions on Test Day and they will probably be considerably easier than this one. Unless you've already mastered all of the 'big' categories (Algebra, Arithmetic, Formulas, Broader Geometry, Ratios, DS, etc.), then this nitpick category really isn't worth the extra time. GMAT assassins aren't born, they're made, Rich
_________________
Contact Rich at: Rich.C@empowergmat.comThe Course Used By GMAT Club Moderators To Earn 750+ souvik101990 Score: 760 Q50 V42 ★★★★★ ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★



eGMAT Representative
Joined: 04 Jan 2015
Posts: 3219

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
21 Apr 2015, 22:43
kshitij89 wrote: Hi Harsh,
Thanks for your reply.
The explanation was crisp and clear but I am not sure if I will be able to solve similar questions with different nos.
Can you please provide examples of similar questions to further test my understanding ?
Regards Kshitij Hi kshitij89  what you need is more practice to get yourself comfortable with the graphical method of solving inequalities. To begin with, I would suggest you to plot lines on the XY coordinate system and find out points which lie on either side of the line. You may further extend this exercise to plotting of 2 lines in the XY coordinate system and finding points which satisfy corresponding inequalities of both the lines. Once you get used to it, you will prefer using the graphical method for solving inequalities. Getting comfortable with the XY coordinate system will also strengthen your understanding of the Coordinate Geometry section For your practice, you may refer the following posts which uses graphical method for solving questions on inequalities and coordinate geometry: inthexyplaneregionrconsistsofallthepointsxy102233.htmlinthexyplaneregionaconsistsofallthepointsxy154784.html?hilit=inequalities%20inequalities%20graphpointxyisapointwithinthetrianglewhatisthe139419.htmlsettconsistsofallpointsxysuchthatx2y21if15626.htmlinthecoordinateplanerectangularregionrhasverticesa104869.htmlHope it helps! Regards Harsh
_________________



Senior Manager
Joined: 10 Mar 2013
Posts: 459
Location: Germany
Concentration: Finance, Entrepreneurship
GPA: 3.88
WE: Information Technology (Consulting)

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
17 Jul 2015, 02:26
1. Rewrite the equation (Y=mx+b) > y=>2/3*x+2 2. Set x=0 and then y=0 > (0, =>2); (3<=, 0) so we have now two points the coordinate plane 3. Draw the line (see attachment) Y you can see that Quadrant IV is not involved there (E)
Attachments
PS123.jpg [ 9.21 KiB  Viewed 15744 times ]



Target Test Prep Representative
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2806

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
06 Jul 2017, 17:33
Quote: In the rectangular coordinate system shown above, which quadrant, if any, contains no point (x,y) that satisfies the inequality \(2x  3y\leq{ 6}\) ? (A) None (B) I (C) II (D) III (E) IV Let’s rewrite the inequality: 3y ≤ 2x  6 y ≥ (2/3)x + 2 We see that the graph of the inequality y ≥ (2/3)x + 2 consists of the line y = (2/3)x + 2, which is a positively sloped line with a yintercept of 2 and the region above this line. Thus, the one quadrant that would not satisfy this inequality is quadrant IV. Answer: E
_________________
5star rated online GMAT quant self study course See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews
If you find one of my posts helpful, please take a moment to click on the "Kudos" button.



Senior SC Moderator
Joined: 22 May 2016
Posts: 3725

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
18 Dec 2017, 01:43
Bunuel wrote: In the rectangular coordinate system shown above, which quadrant, if any, contains no point (x,y) that satisfies the inequality 2x – 3y ≤ –6? (A) None (B) I (C) II (D) III (E) IV Attachment: The attachment 20171218_1011_001.png is no longer available Attachment:
quadrants.png [ 8.25 KiB  Viewed 9544 times ]
Rewrite the inequality in slopeintercept form for the inequality's boundary line\(2x – 3y ≤ –6\) \(3y ≤ 2x  6\) Because dividing by 3, switch the sign: \(y \geq \frac{2}{3} x + 2\) Slope is positive. Positive slope always runs through Quadrants I and III Eliminate answers B and D Find the intercepts to graph the line and/or to assess signsSet x equal to 0 to find yintercept of the inequality's boundary line \(y \geq \frac{2}{3} (0) + 2\) \(y \geq 2\) Set y equal to 0 to find xintercept of boundary line \(0 \geq \frac{2}{3} x + 2\) \(2 \geq \frac{2}{3} x\) \((\frac{3}{2})(2) \geq{x}\) \(x ≤  3\) Intercepts of the inequality's boundary line are (0,2) and (3,0) AssessI graphed the intercepts, connected the points and extended the line. See diagram. No point in Quadrant IV satisfies the inequality You don't have to graph. (I graph often. For me, it is quick. This problem took just under a minute.) Quadrants II has positive yvalues and negative xvalues.  yintercept is positive  xintercept is negative Those signs fit: The boundary line runs through Quadrant II. Eliminate Answer C At this point, we know the graphed inequality's boundary line runs through Quadrants I, II, and III. All straight lines pass through at least two, and at most three, quadrants. (Most straight lines pass through three quadrants. Never four.) By POE, Quadrant IV contains no points that satisfy the inequality. AND/OR Quadrant IV has no positive yvalues and no negative xvalues.  yintercept is positive. The xintercept is negative.  Quadrant IV's (x, y) values are exactly the opposite: (+x, y) No point in Quadrant IV satisfies this inequality Answer
_________________
SC Butler has resumed! Get two SC questions to practice, whose links you can find by date, here.Never doubt that a small group of thoughtful, committed citizens can change the world; indeed, it's the only thing that ever has  Margaret Mead



Director
Joined: 31 Jul 2017
Posts: 503
Location: Malaysia
GPA: 3.95
WE: Consulting (Energy and Utilities)

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
18 Dec 2017, 02:43
genxer123 wrote: Bunuel wrote: In the rectangular coordinate system shown above, which quadrant, if any, contains no point (x,y) that satisfies the inequality 2x – 3y ≤ –6? (A) None (B) I (C) II (D) III (E) IV Attachment: 20171218_1011_001.png Attachment: quadrants.png Rewrite the inequality in slopeintercept form for the inequality's boundary line\(2x – 3y ≤ –6\) \(3y ≤ 2x  6\) Because dividing by 3, switch the sign: \(y \geq \frac{2}{3} x + 2\) Slope is positive. Positive slope always runs through Quadrants I and III Eliminate answers B and D Find the intercepts to graph the line and/or to assess signsSet x equal to 0 to find yintercept of the inequality's boundary line \(y \geq \frac{2}{3} (0) + 2\) \(y \geq 2\) Set y equal to 0 to find xintercept of boundary line \(0 \geq \frac{2}{3} x + 2\) \(2 \geq \frac{2}{3} x\) \((\frac{3}{2})(2) \geq{x}\) \(x ≤  3\) Intercepts of the inequality's boundary line are (0,2) and (3,0) AssessI graphed the intercepts, connected the points and extended the line. See diagram. No point in Quadrant IV satisfies the inequality You don't have to graph. (I graph often. For me, it is quick. This problem took just under a minute.) Quadrants II has positive yvalues and negative xvalues.  yintercept is positive  xintercept is negative Those signs fit: The boundary line runs through Quadrant II. Eliminate Answer C At this point, we know the graphed inequality's boundary line runs through Quadrants I, II, and III. All straight lines pass through at least two, and at most three, quadrants. (Most straight lines pass through three quadrants. Never four.) By POE, Quadrant IV contains no points that satisfy the inequality. AND/OR Quadrant IV has no positive yvalues and no negative xvalues.  yintercept is positive. The xintercept is negative.  Quadrant IV's (x, y) values are exactly the opposite: (+x, y) No point in Quadrant IV satisfies this inequality Answer Hi genxer123, The question asks if any (x,y) of the 4Quadrants will not satisfy \(\frac{x}{3} + \frac{y}{2} ≤ 1\) equation. In IV quadrant (6,4) will definitely satisfy the equation. Can you please clarify my doubt or am I missing something here??



Math Expert
Joined: 02 Aug 2009
Posts: 8335

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
18 Dec 2017, 02:54
rahul16singh28 wrote: Bunuel wrote: In the rectangular coordinate system shown above, which quadrant, if any, contains no point (x,y) that satisfies the inequality 2x – 3y ≤ –6? (A) None (B) I (C) II (D) III (E) IV Attachment: 20171218_1011_001.png Hi genxer123, The question asks if any (x,y) of the 4Quadrants will not satisfy \frac{x}{3 + y/2 < 1} equation. In IV quadrant (6,4) will definitely satisfy the equation. Can you please clarify my doubt or am I missing something here?? HI either draw a quadrant and then find answer OR simply look at equation and find answer.. 2x – 3y ≤ –6.... Right Hand Side is 6 or NEGATIVE.. when will the LHS be POSITIVE.. when 2x is positive or x is positive and 3y is also positive or y is positive means y is negativeso y NEGATIVE and xPOSITIVE THIS is exactly what is there in quad IV so LHS can never be ≤6 E
_________________



Director
Joined: 31 Jul 2017
Posts: 503
Location: Malaysia
GPA: 3.95
WE: Consulting (Energy and Utilities)

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
18 Dec 2017, 02:59
chetan2u wrote: rahul16singh28 wrote: Bunuel wrote: In the rectangular coordinate system shown above, which quadrant, if any, contains no point (x,y) that satisfies the inequality 2x – 3y ≤ –6? (A) None (B) I (C) II (D) III (E) IV Attachment: 20171218_1011_001.png Hi genxer123, The question asks if any (x,y) of the 4Quadrants will not satisfy \frac{x}{3 + y/2 < 1} equation. In IV quadrant (6,4) will definitely satisfy the equation. Can you please clarify my doubt or am I missing something here?? HI either draw a quadrant and then find answer OR simply look at equation and find answer.. 2x – 3y ≤ –6.... Right Hand Side is 6 or NEGATIVE.. when will the LHS be POSITIVE.. when 2x is positive or x is positive and 3y is also positive or y is positive means y is negativeso y NEGATIVE and xPOSITIVE THIS is exactly what is there in quad IV so LHS can never be ≤6 E Hi Chetan2u, Can't the equation 2x – 3y ≤ –6, be written as \(\frac{−x}{3}+\frac{y}{2} ≤ 1\). Apologies if its a stupid question



Math Expert
Joined: 02 Aug 2009
Posts: 8335

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
18 Dec 2017, 03:03
rahul16singh28 wrote: Hi Chetan2u, Can't the equation 2x – 3y ≤ –6, be written as \(\frac{−x}{3}+\frac{y}{2} ≤ 1\). Apologies if its a stupid question Where you have gone wrong \(\frac{x}{3 + y/2 < 1}\) when you divide both sides by 6 , CHANGE the sign2x – 3y ≤ –6 will become \(\frac{2x}{6}\frac{3y}{6}>\frac{6}{6}.......\frac{x}{3}+\frac{y}{2}>1\) a simpler look x<1 will become x>1
_________________



Senior SC Moderator
Joined: 22 May 2016
Posts: 3725

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
18 Dec 2017, 12:31
rahul16singh28 wrote: Bunuel wrote: In the rectangular coordinate system shown above, which quadrant, if any, contains no point (x,y) that satisfies the inequality 2x – 3y ≤ –6? (A) None (B) I (C) II (D) III (E) IV Attachment: 20171218_1011_001.png Hi genxer123, The question asks if any (x,y) of the 4Quadrants will not satisfy \(\frac{x}{3} + \frac{y}{2} ≤ 1\) equation. In IV quadrant (6,4) will definitely satisfy the equation. Can you please clarify my doubt or am I missing something here?? rahul16singh28 , I think chetan2u has answered your question (you simply forgot to switch the sign when you divided by 6). I appreciate your polite and gracious manner.
_________________
SC Butler has resumed! Get two SC questions to practice, whose links you can find by date, here.Never doubt that a small group of thoughtful, committed citizens can change the world; indeed, it's the only thing that ever has  Margaret Mead



Intern
Joined: 08 Dec 2018
Posts: 14

Re: In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
15 Jul 2019, 21:25
Bunuel wrote: SOLUTION
In the rectangular coordinate system shown above, which quadrant, if any, contains no point (x,y) that satisfies the inequality \(2x  3y\leq{ 6}\) ?
(A) None (B) I (C) II (D) III (E) IV
\({2x3y}\leq{6}\) > \(y\geq{\frac{2}{3}x+2}\). Thi inequality represents ALL points, the area, above the line \(y={\frac{2}{3}x+2}\). If you draw this line you'll see that the mentioned area is "above" IV quadrant, does not contains any point of this quadrant.
Else you can notice that if \(x\) is positive, \(y\) can not be negative to satisfy the inequality \(y\geq{\frac{2}{3}x+2}\), so you can not have positive \(x\), negative \(y\). But IV quadrant consists of such \((x,y)\) points for which \(x\) is positive and \(y\) negative. Thus answer must be E.
Answer: E. Can we do it by just checking the slope of the line? As the slope of this line is positive, so the line will not pass from the IV quadrant.



VP
Joined: 14 Feb 2017
Posts: 1364
Location: Australia
Concentration: Technology, Strategy
GMAT 1: 560 Q41 V26 GMAT 2: 550 Q43 V23 GMAT 3: 650 Q47 V33 GMAT 4: 650 Q44 V36 GMAT 5: 650 Q48 V31 GMAT 6: 600 Q38 V35 GMAT 7: 710 Q47 V41
GPA: 3
WE: Management Consulting (Consulting)

In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
Show Tags
18 Jul 2019, 18:03
put the equation in general form and solve \(3y <= 6 2x\) \(y>= 2+\frac{2}{3}*x\) be sure to flip the sign since we divide by a negative or \(y>= \frac{2}{3}*x + 2\) If we have the general form of the line we can plot the line since all we need to know is that the gradient, 2/3, is positive, and that the y intercept is 2. As you can see quadrant IV is the only quadrant that cannot be touched.
Attachments
Capture.JPG [ 22.28 KiB  Viewed 4015 times ]
_________________
Here's how I went from 430 to 710, and how you can do it yourself: https://www.youtube.com/watch?v=KGY5vxqMeYk&t=




In the rectangular coordinate system shown above, which quadrant, if
[#permalink]
18 Jul 2019, 18:03






