Graphing inequalities
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Graphing Inequalities
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Solving and graphing linear inequalities in two variables 1
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Graphing Linear Inequalities in Two Variables Example 2
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Graphing Inequalities 2
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Graphing linear inequalities in two variables 3
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Graphs of inequalities
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Graphing linear inequalities
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Graphing and solving linear inequalities
Graphing Inequalities Graphing Inequalities
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- Let's graph ourselves some inequalities.
- So let's say I had the inequality y is less than or
- equal to 4x plus 3.
- On our xy coordinate plane, we want to show all the x and y
- points that satisfy this condition right here.
- So a good starting point might be to break up this less than
- or equal to, because we know how to graph y is
- equal to 4x plus 3.
- So this thing is the same thing as y could be less than
- 4x plus 3, or y could be equal to 4x plus 3.
- That's what less than or equal means.
- It could be less than or equal.
- And the reason why I did that on this first example problem
- is because we know how to graph that.
- So let's graph that.
- Try to draw a little bit neater than that.
- So that is-- no, that's not good.
- So that is my vertical axis, my y-axis.
- This is my x-axis, right there.
- And then we know the y-intercept, the
- y-intercept is 3.
- So the point 0, 3-- 1, 2, 3-- is on the line.
- And we know we have a slope of 4.
- Which means if we go 1 in the x-direction, we're going to go
- up 4 in the y.
- So 1, 2, 3, 4.
- So it's going to be right here.
- And that's enough to draw a line.
- We could even go back in the x-direction.
- If we go 1 back in the x-direction, we're
- going to go down 4.
- 1, 2, 3, 4.
- So that's also going to be a point on the line.
- So my best attempt at drawing this line is going to look
- something like-- this is the hardest part.
- It's going to look something like that.
- That is a line.
- It should be straight.
- I think you get the idea.
- That right there is the graph of y is equal to 4x plus 3.
- So let's think about what it means to be less than.
- So all of these points satisfy this
- inequality, but we have more.
- This is just these points over here.
- What about all these where y ix less than 4x plus 3?
- So let's think about what this means.
- Let's pick up some values for x.
- When x is equal to 0, what does this say?
- When x is equal to 0, then that means y is going to be
- less than 0 plus 3. y is less than 3.
- When x is equal to negative 1, what is this telling us?
- 4 times negative 1 is negative 4, plus 3 is negative 1. y
- would be less than negative 1.
- When x is equal to 1, what is this telling us?
- 4 times 1 is 4, plus 3 is 7.
- So y is going to be less than 7.
- So let's at least try to plot these.
- So when x is equal to-- let's plot this one first. When x is
- equal to 0, y is less than 3.
- So it's all of these points here-- that I'm shading in in
- green-- satisfy that right there.
- If I were to look at this one over here, when x is negative
- 1, y is less than negative 1.
- So y has to be all of these points down here.
- When x is equal to 1, y is less than 7.
- So it's all of these points down here.
- And in general, you take any point x-- let's say you take
- this point x right there.
- If you evaluate 4x plus 3, you're going to get the point
- on the line.
- That is that x times 4 plus 3.
- Now the y's that satisfy it, it could be equal to that
- point on the line, or it could be less than.
- So it's going to go below the line.
- So if you were to do this for all the possible x's, you
- would not only get all the points on this line which
- we've drawn, you would get all the points below the line.
- So now we have graphed this inequality.
- It's essentially this line, 4x plus 3, with all of the area
- below it shaded.
- Now, if this was just a less than, not less than or equal
- sign, we would not include the actual line.
- And the convention to do that is to actually make the line a
- dashed line.
- This is the situation if we were dealing with just less
- than 4x plus 3.
- Because in that situation, this wouldn't apply, and we
- would just have that.
- So the line itself wouldn't have satisfied it, just the
- area below it.
- Let's do one like that.
- So let's say we have y is greater than negative x
- over 2 minus 6.
- So a good way to start-- the way I like to start these
- problems-- is to just graph this equation right here.
- So let me just graph-- just for fun-- let me graph y is
- equal to-- this is the same thing as negative 1/2 minus 6.
- So if we were to graph it, that is my vertical axis, that
- is my horizontal axis.
- And our y-intercept is negative 6.
- So 1, 2, 3, 4, 5, 6.
- So that's my y-intercept.
- And my slope is negative 1/2.
- Oh, that should be an x there, negative 1/2 x minus 6.
- So my slope is negative 1/2, which means when I go 2 to the
- right, I go down 1.
- So if I go 2 to the right, I'm going to go down 1.
- If I go 2 to the left, if I go negative 2, I'm
- going to go up 1.
- So negative 2, up 1.
- So my line is going to look like this.
- My line is going to look like that.
- That's my best attempt at drawing the line.
- So that's the line of y is equal to
- negative 1/2 x minus 6.
- Now, our inequality is not greater than or equal, it's
- just greater than negative x over 2 minus 6, or greater
- than negative 1/2 x minus 6.
- So using the same logic as before, for any x-- so if you
- take any x, let's say that's our particular x we want to
- pick-- if you evaluate negative x over 2 minus 6,
- you're going to get that point right there.
- You're going to get the point on the line.
- But the y's that satisfy this inequality are the y's
- greater than that.
- So it's going to be not that point-- in fact, you draw an
- open circle there-- because you can't include the point of
- negative 1/2 x minus 6.
- But it's going to be all the y's greater than that.
- That'd be true for any x.
- You take this x.
- You evaluate negative 1/2 or negative x over 2 minus 6,
- you're going to get this point over here.
- The y's that satisfy it are all the y's above that.
- So all of the y's that satisfy this equation, or all of the
- coordinates that satisfy this equation, is this entire area
- above the line.
- And we're not going to include the line.
- So the convention is to make this line into a dashed line.
- And let me draw-- I'm trying my best to turn it into a
- dashed line.
- I'll just erase sections of the line, and hopefully it
- will look dashed to you.
- So I'm turning that solid line into a dashed line to show
- that it's just a boundary, but it's not included in the
- coordinates that satisfy our inequality.
- The coordinates that satisfy our equality are all of this
- yellow stuff that I'm shading above the line.
- Anyway, hopefully you found that helpful.
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