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Question 204541: (3) When graphing a linear inequality, how do you know if the inequality represents the area above the line? Give examples.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you follow the inequality.
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assume y < x
you graph y = x.
the area under the graph of the equation y = x is the area you are interested in.
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assume y < -x
you graph y = -x
the area under the graph of the equation y = -x is the area you are interested in.
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you have two equations or constraints.
y < x
y < -x
these are 2 intersecting lines that are perpendicular to each other.
you graph y = x
you graph y = -x
the area under the graph of the equation y = x and the graph of the equation y = -x is the area you are interested in.
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suppose you have 3 equations or constraints.
y < x
y < -x
y > -3
you graph y = x
you graph y = -x
you graph y = -3
the area you are interested is the area under the graph of the equation y = x and the area under the graph of the equation y = -x and the area above the graph of the equation y = -3
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a graph of the 3 equations would look like this:
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you can see on this graph that if you were only concerned with y < x, then any value of y under the graph of the equation y = x would be good.
if x = -4, then y < -4 would satisfy the inequality.
if x = 4, then y < 4 would satisfy the inequality.
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you can see on this graph that if you were only concerned with y < -x, then any value of y under the graph of the equation y = -x would be good.
if x = -4, then y < 4 would satisfy the inequality
if x = 4, then y < -4 would satisfy the inequality.
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you can see on this graph that if you were concerned with y < x and y < -x, then any value of y under the graph of the equation y = x and under the graph of the equation y = -x would be good.
if x = -4, then y would have to be smaller than -4
if x = 4, then y would have to be smaller than -4 again.
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you can see on this graph that if you were concerned with y < x and y < -x and y > -3, then any value of y under the graph of the equation y = x and under the graph of the equation y = -x and above the graph of y = -3 would be good.
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if x = -4, then there would be no y values that would satisfy the inequality because y could not be below -4 and above -3 at the same time.
if x = 4, then there would be no y values that would satisfy the inequality because y could not be below -4 again and above -3 at the same time.
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x would have to be between -3 and 3 and then y would have to be below the lines y = -x and y = x and above the line y = -3.
if x was 2, then y would have to be below -2 and above -3.
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to spot the area where you want to be sometimes it's necessary to take some points to verify what you are seeing is what satisfies all equations in the inequality.
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if the inequality is <, than the area of interest is below the graph of the equation.
if the inequality is >, than the area of interest is above the graph of the equation.
the graph of the equation is the equality.
example:
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y = x^2 + 3x + 5 is the equality that is used to graph this equation.
y < x^2 + 3x + 5 is the area under the graph of this equation.
y > x^2 + 3x + 5 is the area above the graph of this equation.
the graph looks like this:
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pick any x value on this graph.
try (-2)
(-2)^2 + 3*(-2) + 5 = 4 - 6 + 5 = 3
when x = -2, y = 3 should be on the graph of the equation.
It is as shown by the intersection of the vertical line at x = -2 and the horizontal line at y = 3.
when y > 3 you can see it's in the area of the graph above the equation.
when y < 3 you can see it's in the area of the graph below the equation.
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