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

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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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.

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.

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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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

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

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!

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)

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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