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

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All points (x,y) that lie below the line l, shown above, satisfy which [#permalink] New post   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. \(y < -\frac{1}{2}*x + 3\)


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E

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.
Edited the question.
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Re: All points (x,y) that lie below the line l, shown above, satisfy which [#permalink] New post   13 Oct 2009, 12:28
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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 < 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.

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 [#permalink] New post   30 Jun 2010, 02:39
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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.
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Re: All points (x,y) that lie below the line l, shown above, satisfy which [#permalink] New post   14 Oct 2009, 04:55
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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 [#permalink] New post   28 Jun 2013, 01:30
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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.
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Re: All points (x,y) that lie below the line l, shown above, satisfy which [#permalink] New post   28 Jun 2013, 01:45
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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 [#permalink] New post   28 Jun 2013, 01:55
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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
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Re: All points (x,y) that lie below the line l, shown above, satisfy which [#permalink] New post   30 Jun 2013, 02:59
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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!
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Re: All points (x,y) that lie below the line l, shown above, satisfy which [#permalink] New post   20 Oct 2013, 03:49
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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 [#permalink] New post   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
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Re: All points (x,y) that lie below the line l, shown above, satisfy which [#permalink]
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