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All points (x,y) that lie below the line l, shown above, satisfy which
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Updated on: 10 Oct 2019, 05:07
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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. \(y < \frac{1}{2}*x + 3\) E. \(y < \frac{1}{2}*x + 3\) Attachment:
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Originally posted by study on 13 Oct 2009, 12:08.
Last edited by Bunuel on 10 Oct 2019, 05:07, edited 3 times in total.
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Re: All points (x,y) that lie below the line l, shown above, satisfy which
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13 Oct 2009, 12:28
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 \(\frac{yy_1}{xx_1}=\frac{y_1y_2}{x_1x_2}\)So equation of a line which passes the points A(0,3) and B(6,0) would be: \(\frac{y3}{x0}=\frac{30}{06}\) > \(2y+x6=0\) > \(y=\frac{1}{2}x+3\) Points below this line satisfy the inequality: \(y<\frac{1}{2}x+3\) ORThe 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 yintercept 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 \(m=\frac{y_1y_2}{x_1x_2}\) >\(\frac{30}{06}=\frac{1}{2}\) 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. Answer: E. For more please check Coordinate Geometry chapter of the Math Book (link in my signature). Hope it's clear.
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Re: All points (x,y) that lie below the line l, shown above, satisfy which
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30 Jun 2010, 02:39
Hi,
compute the slope. The slope is the change in rise over run or (y2y1)/(x2x1). So, slope is (30)/(06) = 1/2. (Or, (03)/(60) = 1/2. And the yintercept of the line (where the line crosses or touches the yaxis) 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 yintercept), 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.




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Re: All points (x,y) that lie below the line l, shown above, satisfy which
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14 Oct 2009, 04:55
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.



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Re: All points (x,y) that lie below the line l, shown above, satisfy which
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28 Jun 2013, 01:30
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 \(\frac {x} {xintercept} + \frac {y} {yintercept} 1 = 0\) 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+01 > 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.
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Re: All points (x,y) that lie below the line l, shown above, satisfy which
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28 Jun 2013, 01:45
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



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Re: All points (x,y) that lie below the line l, shown above, satisfy which
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28 Jun 2013, 01:55
Bunuel wrote: Bumping for review and further discussion*. Get a kudos point for an alternative solution! *New project from GMAT Club!!! Check HEREWe 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
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Re: All points (x,y) that lie below the line l, shown above, satisfy which
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30 Jun 2013, 02:59
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 plugin 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!



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Re: All points (x,y) that lie below the line l, shown above, satisfy which
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20 Oct 2013, 03:49
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)
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Re: All points (x,y) that lie below the line l, shown above, satisfy which
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19 Jul 2019, 19:52
Find the equation of the line by using two given points (0,3) and (6,0). Find slope (03)/(60) =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
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Re: All points (x,y) that lie below the line l, shown above, satisfy which
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10 Oct 2019, 05:04
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Re: All points (x,y) that lie below the line l, shown above, satisfy which
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