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Let's graph ourselves some inequalities.

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So let's say I had the inequality y is less than or

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equal to 4x plus 3.

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On our xy coordinate plane, we want to show all the x and y

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points that satisfy this condition right here.

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So a good starting point might be to break up this less than

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or equal to, because we know how to graph y is

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equal to 4x plus 3.

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So this thing is the same thing as y could be less than

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4x plus 3, or y could be equal to 4x plus 3.

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That's what less than or equal means.

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It could be less than or equal.

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And the reason why I did that on this first example problem

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is because we know how to graph that.

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So let's graph that.

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Try to draw a little bit neater than that.

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So that is-- no, that's not good.

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So that is my vertical axis, my y-axis.

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This is my x-axis, right there.

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And then we know the y-intercept, the

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y-intercept is 3.

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So the point 0, 3-- 1, 2, 3-- is on the line.

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And we know we have a slope of 4.

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Which means if we go 1 in the x-direction, we're going to go

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up 4 in the y.

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So 1, 2, 3, 4.

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So it's going to be right here.

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And that's enough to draw a line.

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We could even go back in the x-direction.

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If we go 1 back in the x-direction, we're

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going to go down 4.

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1, 2, 3, 4.

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So that's also going to be a point on the line.

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So my best attempt at drawing this line is going to look

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something like-- this is the hardest part.

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It's going to look something like that.

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That is a line.

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It should be straight.

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I think you get the idea.

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That right there is the graph of y is equal to 4x plus 3.

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So let's think about what it means to be less than.

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So all of these points satisfy this

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inequality, but we have more.

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This is just these points over here.

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What about all these where y ix less than 4x plus 3?

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So let's think about what this means.

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Let's pick up some values for x.

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When x is equal to 0, what does this say?

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When x is equal to 0, then that means y is going to be

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less than 0 plus 3. y is less than 3.

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When x is equal to negative 1, what is this telling us?

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4 times negative 1 is negative 4, plus 3 is negative 1. y

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would be less than negative 1.

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When x is equal to 1, what is this telling us?

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4 times 1 is 4, plus 3 is 7.

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So y is going to be less than 7.

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So let's at least try to plot these.

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So when x is equal to-- let's plot this one first. When x is

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equal to 0, y is less than 3.

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So it's all of these points here-- that I'm shading in in

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green-- satisfy that right there.

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If I were to look at this one over here, when x is negative

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1, y is less than negative 1.

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So y has to be all of these points down here.

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When x is equal to 1, y is less than 7.

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So it's all of these points down here.

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And in general, you take any point x-- let's say you take

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this point x right there.

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If you evaluate 4x plus 3, you're going to get the point

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on the line.

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That is that x times 4 plus 3.

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Now the y's that satisfy it, it could be equal to that

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point on the line, or it could be less than.

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So it's going to go below the line.

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So if you were to do this for all the possible x's, you

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would not only get all the points on this line which

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we've drawn, you would get all the points below the line.

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So now we have graphed this inequality.

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It's essentially this line, 4x plus 3, with all of the area

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below it shaded.

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Now, if this was just a less than, not less than or equal

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sign, we would not include the actual line.

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And the convention to do that is to actually make the line a

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dashed line.

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This is the situation if we were dealing with just less

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than 4x plus 3.

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Because in that situation, this wouldn't apply, and we

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would just have that.

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So the line itself wouldn't have satisfied it, just the

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area below it.

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Let's do one like that.

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So let's say we have y is greater than negative x

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over 2 minus 6.

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So a good way to start-- the way I like to start these

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problems-- is to just graph this equation right here.

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So let me just graph-- just for fun-- let me graph y is

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equal to-- this is the same thing as negative 1/2 minus 6.

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So if we were to graph it, that is my vertical axis, that

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is my horizontal axis.

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And our y-intercept is negative 6.

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So 1, 2, 3, 4, 5, 6.

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So that's my y-intercept.

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And my slope is negative 1/2.

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Oh, that should be an x there, negative 1/2 x minus 6.

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So my slope is negative 1/2, which means when I go 2 to the

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right, I go down 1.

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So if I go 2 to the right, I'm going to go down 1.

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If I go 2 to the left, if I go negative 2, I'm

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going to go up 1.

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So negative 2, up 1.

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So my line is going to look like this.

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My line is going to look like that.

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That's my best attempt at drawing the line.

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So that's the line of y is equal to

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negative 1/2 x minus 6.

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Now, our inequality is not greater than or equal, it's

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just greater than negative x over 2 minus 6, or greater

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than negative 1/2 x minus 6.

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So using the same logic as before, for any x-- so if you

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take any x, let's say that's our particular x we want to

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pick-- if you evaluate negative x over 2 minus 6,

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you're going to get that point right there.

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You're going to get the point on the line.

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But the y's that satisfy this inequality are the y's

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greater than that.

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So it's going to be not that point-- in fact, you draw an

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open circle there-- because you can't include the point of

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negative 1/2 x minus 6.

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But it's going to be all the y's greater than that.

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That'd be true for any x.

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You take this x.

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You evaluate negative 1/2 or negative x over 2 minus 6,

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you're going to get this point over here.

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The y's that satisfy it are all the y's above that.

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So all of the y's that satisfy this equation, or all of the

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coordinates that satisfy this equation, is this entire area

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above the line.

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And we're not going to include the line.

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So the convention is to make this line into a dashed line.

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And let me draw-- I'm trying my best to turn it into a

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dashed line.

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I'll just erase sections of the line, and hopefully it

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will look dashed to you.

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So I'm turning that solid line into a dashed line to show

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that it's just a boundary, but it's not included in the

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coordinates that satisfy our inequality.

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The coordinates that satisfy our equality are all of this

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yellow stuff that I'm shading above the line.

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Anyway, hopefully you found that helpful.