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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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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.
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.
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.
