GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 23 Aug 2019, 09:07

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

All points (x,y) that lie below the line l, shown above

Author Message
TAGS:

Hide Tags

Manager
Joined: 05 Oct 2008
Posts: 235
All points (x,y) that lie below the line l, shown above  [#permalink]

Show Tags

Updated on: 28 Jun 2013, 01:35
1
23
00:00

Difficulty:

35% (medium)

Question Stats:

67% (01:34) correct 33% (02:07) wrong based on 647 sessions

HideShow timer Statistics

Attachment:

Line.png [ 10.1 KiB | Viewed 19520 times ]
All points (x,y) that lie below the line l, shown above, satisfy which of the following inequalities?

A. y < 2x + 3
B. y < -2x + 3
C. y < -x + 3
D. y < 1/2*x + 3
E. y < -1/2*x + 3

Originally posted by study on 13 Oct 2009, 12:08.
Last edited by Bunuel on 28 Jun 2013, 01:35, edited 1 time in total.
Edited the question.
Math Expert
Joined: 02 Sep 2009
Posts: 57279

Show Tags

13 Oct 2009, 12:28
15
3

All points (x,y) that lie below the line l, shown above, satisfy which of the following inequalities?
A. y < 2x + 3
B. y < -2x + 3
C. y < -x + 3
D. y < 1/2*x + 3
E. y < -1/2*x + 3

First of all we should write the equation of the line $$l$$:

We have two points: A(0,3) and B(6,0).

Equation of a line which passes through two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$ is $$\frac{y-y_1}{x-x_1}=\frac{y_1-y_2}{x_1-x_2}$$

So equation of a line which passes the points A(0,3) and B(6,0) would be: $$\frac{y-3}{x-0}=\frac{3-0}{0-6}$$ --> $$2y+x-6=0$$ --> $$y=-\frac{1}{2}x+3$$

Points below this line satisfy the inequality: $$y<-\frac{1}{2}x+3$$

OR
The equation of line which passes through the points $$A(0,3)$$ and $$B(6,0)$$ can be written in the following way:

Equation of a line in point intercept form is $$y=mx+b$$, where: $$m$$ is the slope of the line and $$b$$ is the y-intercept of the line (the value of $$y$$ for $$x=0$$).

The slope of a line, $$m$$, is the ratio of the "rise" divided by the "run" between two points on a line, thus $$m=\frac{y_1-y_2}{x_1-x_2}$$ -->$$\frac{3-0}{0-6}=-\frac{1}{2}$$ and $$b$$ is the value of $$y$$ when $$x=0$$ --> A(0,3) --> $$b=3$$.

So the equation is $$y=-\frac{1}{2}x+3$$

Points below this line satisfy the inequality: $$y<-\frac{1}{2}x+3$$.

Actually one could guess that the answer is E at the stage of calculating the slope $$m=-\frac{1}{2}$$, as only answer choice E has the same slope line in it.

For more please check Coordinate Geometry chapter of the Math Book (link in my signature).

Hope it's clear.
_________________
Kaplan GMAT Instructor
Joined: 21 Jun 2010
Posts: 69
Location: Toronto
Re: gmat prep,, PS question  [#permalink]

Show Tags

30 Jun 2010, 02:39
5
1
Hi,

compute the slope. The slope is the change in rise over run or (y2-y1)/(x2-x1). So, slope is (3-0)/(0-6) = -1/2. (Or, (0-3)/(6-0) = -1/2. And the y-intercept of the line (where the line crosses or touches the y-axis) is +3. Thus, using the line equation y = mx + b (in which "y" and "x" is an ordinate pair for any point on the line, "m" is slope, and "b" is y-intercept), the equation of the line is y = -1/2x + 3. We're looking for points that lie below this line, so for any given value of x, the y value should be the biggest possible value without going above the line. Thus, the correct answer is choice E.

You can pick numbers to confirm. Let x = 6. We know that when x = 6, according to the line, y = 0. Let's plug x = 6 into the line equation: y = (-1/2)*6 + 3 = 0. Yep, that's right. To fall below the line, then, when x = 6, y<0.

If you understand the line equation, then as soon as you computed the slope of "-1/2", you know that the answer is E, and you're done (because none of the other choices have "-1/2").

We could have also observed that the slope is negative (because the line is "falling" reading from left to right). That observation allows us to cancel choices A and D. The slope is definitely not -1...eliminate C. Choice B is a trap for someone who reversed the given and x and y coordinates or else a trap for someone who computed slope as run/rise rather than rise/run.
General Discussion
Senior Manager
Joined: 31 Aug 2009
Posts: 364
Location: Sydney, Australia

Show Tags

14 Oct 2009, 04:55
4
The line will be in the form of y=mx+b
In this case we are looking for a negative m (eliminate option A and D).
Finally we are looking for a line with X intercept of 6.
So if you make y=0 for all remaining formulas you get

b) x=3/2 wrong
c) x=3 wrong
e) x=6 RIGHT

ANS = E.
Intern
Joined: 22 Dec 2009
Posts: 38

Show Tags

30 Jun 2010, 01:40
Can someone help me solve this problem please!!!

All points (x,y) that lie below the line L, shown above, satisfy which of the following inequalities..

*I couldnt draw the xy-plane but, y is +3 and x is +6 and line L is lined from y to x straight..

1) y<2x+3
2) y<-2x+3
3) y<-x+3
4) y<1/2x+3
5) y<-1/2x+3

thanks
Intern
Joined: 22 Dec 2009
Posts: 38

Show Tags

30 Jun 2010, 03:15

I understood now and you are right I didnt click to think this line going down from left to right which creates negative slope...

thank you so much!!
Kaplan GMAT Instructor
Joined: 21 Jun 2010
Posts: 69
Location: Toronto

Show Tags

30 Jun 2010, 13:53
gmatJP wrote:

I understood now and you are right I didnt click to think this line going down from left to right which creates negative slope...

thank you so much!!

You're most welcome!
Manager
Joined: 17 Mar 2010
Posts: 137

Show Tags

04 Sep 2010, 23:55
Looks difficult but after solving, becomes eazy.... good.
Math Expert
Joined: 02 Sep 2009
Posts: 57279
Re: All points (x,y) that lie below the line l, satisfy which of  [#permalink]

Show Tags

27 Jun 2013, 23:45
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Coordinate Geometry: math-coordinate-geometry-87652.html

All DS Coordinate Geometry Problems to practice: search.php?search_id=tag&tag_id=41
All PS Coordinate Geometry Problems to practice: search.php?search_id=tag&tag_id=62

_________________
Verbal Forum Moderator
Joined: 10 Oct 2012
Posts: 603
Re: All points (x,y) that lie below the line l, satisfy which of  [#permalink]

Show Tags

28 Jun 2013, 01:30
2
study wrote:
All points (x,y) that lie below the line l, satisfy which of the following inequalities?

A. y<2x+3
B. y<-2x+3
C. y<-x+3
D. y<1/2x+3
E. y<-1/2x+3
Attachment:
new_PS_Lines_E.JPG

For any given line L, if the intercepts are given, we can write the equation of the line as $$\frac {x} {x-intercept} + \frac {y} {y-intercept} -1 = 0$$

For the given line, it stands as L = $$\frac {x} {6} + \frac {y} {3} -1 = 0$$ Now, notice that when the value of origin is plugged in (0,0), we get L as 0+0-1 --> L<0. Thus, the origin lies on the negative side of the given line. And, as origin lies below the given line, all the points in that region will make L<0 -->

$$\frac {x} {6} + \frac {y} {3} -1<0$$ --> $$\frac {y} {3}< 1-\frac {x} {6}$$ --> $$y < 3-\frac {x} {2}$$

E.
_________________
Retired Moderator
Joined: 16 Jun 2012
Posts: 999
Location: United States
Re: All points (x,y) that lie below the line l, shown above  [#permalink]

Show Tags

28 Jun 2013, 01:45
1
study wrote:
Attachment:
Line.png
All points (x,y) that lie below the line l, shown above, satisfy which of the following inequalities?

A. y < 2x + 3
B. y < -2x + 3
C. y < -x + 3
D. y < 1/2*x + 3
E. y < -1/2*x + 3

Frankly, I did this question without any calculation. I hope my approach helps you save time.

First step:
We have equation: $$y = ax + b$$ in which a is the slope of the line.
I see the line "l" passes through quadrant II and IV ==> The slope of line "l" should be negative
==> A, D are out immediately.

Second step.
We see two points, say A (6, 0) and B (0, 3) on line "l".
Let plug in one point, say A (6,0) to B, C, E ==> C is out
Let plug in the second point, say B (0,3) to D & E ==> D is out

Only E remains and is correct.

Hope it helps.

PS: You can save a lot of time by using "plug in" method
_________________
Please +1 KUDO if my post helps. Thank you.

"Designing cars consumes you; it has a hold on your spirit which is incredibly powerful. It's not something you can do part time, you have do it with all your heart and soul or you're going to get it wrong."

Chris Bangle - Former BMW Chief of Design.
MBA Section Director
Affiliations: GMAT Club
Joined: 22 Feb 2012
Posts: 6663
City: Pune
Re: All points (x,y) that lie below the line l, satisfy which of  [#permalink]

Show Tags

28 Jun 2013, 01:55
1
Bunuel wrote:
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

We have X intercept as 6 and Y intercept as 3

The equation of a line having a as X intercept and b as Y intercept is $$\frac{x}{a}$$ + $$\frac{y}{b}$$ = 1

So the equation of the line above would be $$\frac{x}{6}$$ + $$\frac{y}{3}$$ = 1 ----------> Y = $$\frac{-1x}{2}$$ + 3

All the points below the line would satisfy the inequality Y < $$\frac{-1x}{2}$$ + 3

Hence Choice E
_________________
2020 MBA Applicants: Introduce Yourself Here!

MBA Video Series - Video answers to specific components and questions about MBA applications.

2020 MBA Deadlines, Essay Questions and Analysis of all top MBA programs
Manager
Joined: 28 Feb 2012
Posts: 106
GPA: 3.9
WE: Marketing (Other)
Re: All points (x,y) that lie below the line l, shown above  [#permalink]

Show Tags

30 Jun 2013, 02:59
1
study wrote:
Attachment:
Line.png
All points (x,y) that lie below the line l, shown above, satisfy which of the following inequalities?

A. y < 2x + 3
B. y < -2x + 3
C. y < -x + 3
D. y < 1/2*x + 3
E. y < -1/2*x + 3

As an alternative solution to this question i suggest to plug-in 0 for both x and y to find the x and y intercepts. From the graph it is clearly seen that the values of y should be less than 3 and the values of x should be less than 6 so ideally when we find the x and y intercepts should get the y<3 and x<6.

a) x=0 then y<3 this part works; y=0, x>-1,5 not our target;
b) x=0 then y<3 this part works; y=0, x<1,5 not our target;
c) x=0 then y<3 this part works; y=0, x<3 not our target;
d) x=0 then y<3 this part works; y=0, x>-6 not our target;
e) x=0 then y<3 this part works; y=0, x<6 BINGO!

E is the line in the graph satisfies y< -1/2*x + 3. This method seems timeconsuming but for those who forget the functions of slope and othe formulas this is very basic visual solution. It took just under 2 min, plus just from one glance it is seen that y<3 in all options so no need to spend time for y, just concentrate to find option which satisfies for x.

Hope that helps!
_________________
If you found my post useful and/or interesting - you are welcome to give kudos!
Senior Manager
Joined: 02 Mar 2012
Posts: 283
Schools: Schulich '16
Re: All points (x,y) that lie below the line l, shown above  [#permalink]

Show Tags

23 Aug 2013, 09:59
i just calculated the intercepts for x and y.onle E suffice and slope match
Manager
Joined: 07 May 2013
Posts: 93
Re: All points (x,y) that lie below the line l, shown above  [#permalink]

Show Tags

13 Oct 2013, 02:19
All points (x,y) that lie below the line l, shown above, satisfy which of the following inequalities?

A. y < 2x + 3
B. y < -2x + 3
C. y < -x + 3
D. y < 1/2*x + 3
E. y < -1/2*x + 3

The easiest approach.
Firstly, you have to know that a line which slopes downwards from left to right ALWAYS has -ve slope
=====>options A and D are eliminated
WHY?
ALL equations are of form y=mx+c, where m is slope.

Now, check y-intercept i.e., c in the options B,C and E
ALL are 3 and in the diagram also y-intercept is 3
So, at this point you cannot eliminate any options on basis of y-intercept\

BUT note that in the options slopes for all 3 options is DIFFERENT-this is what you should attack.
HOW?
Observe diagram: two co-ordinates on the line are (0,3) and (6,0)
slope=0-3/6-0=1/2
As line is -ve sloped(discussed earlier) m=-1/2
Only one options exists with this slope and its E.
Intern
Joined: 15 Dec 2007
Posts: 12
Re: All points (x,y) that lie below the line l, shown above  [#permalink]

Show Tags

20 Oct 2013, 03:49
2
Took me 5 seconds to figure out the answer:
When you look at this graph, you can write it right away in this form: y=mx+b
b=point on y coordinate
m=negative > decreasing
m=positive > increasing
m=-1/2 = the line goes from b point to (2,2), as the next y,x integer cross, and (4,1) as the next, and (6,0) next (basically 1 in slope means that it goes 1 down, and 2 means it goes 2 down (if slope is negative, in positive one 1 means one up, 2 means 2 right).

So just by looking at (6,0) point and b=3 you can firmly say that the line is gonna be y=-1/2x+3.

So for instance if you draw line with b=2, and slope 5/3, you will get this green line (y=5/3x+2), and if you draw line with b=2 and slope -5/3, you will get orange line (y=-5/3x+2)
Attachments

File comment: question

2222.png [ 86.3 KiB | Viewed 15316 times ]

Director
Joined: 14 Feb 2017
Posts: 937
Location: Australia
Concentration: Technology, Strategy
Schools: LBS '22
GMAT 1: 560 Q41 V26
GMAT 2: 550 Q43 V23
GMAT 3: 650 Q47 V33
GMAT 4: 650 Q44 V36
WE: Management Consulting (Consulting)
Re: All points (x,y) that lie below the line l, shown above  [#permalink]

Show Tags

19 Jul 2019, 19:52
Find the equation of the line by using two given points (0,3) and (6,0).

Find slope
(0-3)/(6-0)
=-1/2

Find y intercept (when x=0): y = 3 (positive)
Thus equation is y= -x/2 + 3

Since we are finding points less than the line we say y <-x/2 + 3
_________________
Goal: Q49, V41

+1 Kudos if you like my post pls!
Re: All points (x,y) that lie below the line l, shown above   [#permalink] 19 Jul 2019, 19:52
Display posts from previous: Sort by