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We're on problem 21.

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Stan's solution to an equation is shown below.

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All right, and I haven't looked at it yet, but let's

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just see what they ask us.

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Which statement about Stan's solution is true?

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So let's just work through it and see if he got it right or

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if he made a mistake.

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So let's see.

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Let's think about it how we would do it.

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So if we had n plus 8 times n plus 20 is equal to 110.

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So the first thing you want to do is you'd want to distribute

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this 8 times n plus 20.

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You don't want to multiply this whole n plus 8.

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Remember, you're doing order of operations, so it's

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multiplication first. Let's do that.

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So you get the n plus and you distribute this 8.

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So it's 8n plus 160 is equal to 110.

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8 times n plus 8 times 20 is 160.

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And already I see a discrepancy

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between ours and his.

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We have n there.

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We have plus 8n there.

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But where he has a 20, we have 160.

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He didn't distribute the 8.

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8 times n plus 20 is 8n plus 160.

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So that's wrong.

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So he made a mistake in step 1.

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Unfortunately, he got all that other work that he had to do

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just because of his-- it's all going to be wrong because of

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his mistake.

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Problem 22.

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When is the statement true?

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The opposite of a number is less then the original number.

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Well, I mean, if I have a negative number, the opposite

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of a negative is positive.

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So it's not going to be less.

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But if l positive number, then the opposite of it is

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negative, which is less than it.

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Because a negative is less than a positive.

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Or if we say that x is greater than 0, then minus x is going

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to be less than x.

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If we say x is positive, then definitely.

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You know, minus 3 is less than positive 3.

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So the statement is true for positive numbers.

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So that's C.

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It's definitely not never true.

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I mean we showed a case where it's true.

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If I have positive 3, negative 3 is less than that.

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Statement B.

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The statement is always true.

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Well no, if we start with negative 3, the opposite is

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going to be larger.

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The opposite's going to be positive 3.

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And D.

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The statement is true for negative numbers.

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No, it's not.

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Negative 3, the opposite is 3.

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And that's larger than negative 3.

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Problem 23.

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What is the y-intercept of the graph?

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OK, let me see.

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So 23.

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They say 4x plus 2y is equal to 12.

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So if you were to graph this line, not that I'm going to--

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the y-intercept is when it intersects the y-axis.

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So it's that point there.

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It's what's this y-coordinate.

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Well the x-coordinate, it's when x is equal to 0.

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So it's going to be 0 comma some y-intercept.

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So the easiest thing to do is say when x is equal

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to 0, what is y?

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So when x is equal to 0, this 4 times 0 is equal to 0.

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So that just becomes 0.

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So you get 2y is equal to 12.

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y is equal to 6, and that's choice C.

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All they want to know is, of this line, when x is equal to

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0, what's y?

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That's what the y-intercept is.

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Next problem.

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OK, they've drawn us a graph.

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They say-- let me copy and paste this one.

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OK.

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Copied it.

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OK, which inequality is shown on the graph below?

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So first of all, let's figure out the equation of

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this line right now.

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So what's its y-intercept, first of all?

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So when x is equal to 0, y is equal to minus 1.

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If we put it in slope y intercept form, the equation

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of really, any line is y is equal to mx plus b.

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Where m is the slope and b is the y-intercept.

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Here, when x is 0, y is negative 1.

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So the y-intercept is minus 1.

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So b is minus 1.

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So we know that y is equal to mx minus 1.

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And now we just have to figure out the slope.

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And the slope is just change in y for a given change in x.

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So if we say change in y for a given change in x is equal

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to-- let's see.

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When I increase x by 2, I increase y by 1.

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So change in y is 1 when the change in x is 2.

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And you could say when you increase x by 4-- I'm going

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four spots-- I increase y by 2.

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So you could have done that.

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You could have said change in y over change in x is 2/4,

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which is also equal to 1/2.

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But either way, we now know the slope.

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The slope is change in y over change in x.

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So the equation of the line is y is equal to 1/2 x minus 1.

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That's the equation of this line.

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Now they want to know what the equation of the inequality is.

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So first of all, they're including-- they have this

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grey area and they put this line in bold.

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So that means that we're including the line.

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If they had drawn a dotted line here that means we're not

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including the line.

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But since they're including the line, it's going to be

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either greater than or less than or equal to.

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That equal to is because we're including the line.

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But just think about it.

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So our choices are y is greater than or equal to 1/2 x

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minus 1 or y is less than or equal to 1/2 x minus 1.

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Now think about for any given x.

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Let's say when x is equal to 2.

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When x is equal to 2, if you put it into this equation you

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get y is equal to 0 and that's this point on the line.

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Now is the grey area all y values greater than 0 or all y

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values less than 0?

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Well clearly, it's all y values greater than 0.

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It's all the y values above the y value

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dictated by this equation.

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The equation of this inequality or this area is y

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is greater than 1/2 x minus 1.

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And just a very easy way to eyeball it is OK, they're

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including the line, so I'm going to have this equal

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then-- it's going to either be greater than or equal to or

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less than or equal to, and since it's the area above the

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line it's going to be greater than or equal to.

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And that is choice D.

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Next problem, problem 25.

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This is another one where I think it makes sense to copy

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and paste it.

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It's a big one.

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OK.

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Which best represents the graph of y id

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equal to 2x minus 2.

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So the slope is 2.

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Its positive 2 and its y-intercept is minus 2.

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So the y-intercept is easy to eyeball.

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Here the y-intercept is minus 2.

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That's right.

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Here it's minus 2 as well.

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In both of these, the y-intercept is plus 2.

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So neither of these are going to, right?

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It intersects the y-axis at positive 2.

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But we know that the y-intercept is minus 2.

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So we know-- whoops.

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Where'd I go?

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There we go.

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So we know our choices are one of the top two.

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It's either A or C.

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And which of these has a positive slope,

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positive slope of 2?

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Well in choice A when I go to the right by 2, I go up by 4.

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So change in y over change in x.

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I go up by 4, when I go up in x by 2.

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I go up in y by 4 when I go up in x by 2, so the slope is 2.

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So this has a positive slope of 2, so that's choice A.

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But just for seeing why choice C doesn't work or figuring out

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the slope, think about it this way.

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Let me start at a random point here.

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If I increase x value by 2, what's

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happening to my y value?

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Is it increasing by 2 or is it decreasing by 2?

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Well it's decreasing.

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Well it's decreasing by 4.

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It's decreasing at twice the rate.

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It's decreasing by 4.

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So in this case, change in y over change in x.

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Whenever x is positive 2, we increased by 2 here.

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What happened to y.

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It went down 4.

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So here the slope is minus 2.

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And you can look at that.

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You can kind of eyeball it because it's going from the

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top left to the bottom right, so it's going to

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be a negative number.

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But the easiest way I always do is to draw these arrows and

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say, OK, when I increase x, y is decreasing.

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So that's going to be a negative slope.

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Let's do it.

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So the answer was A.

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Next problem.

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OK, so they're giving us another one of these where

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they shaded a graph.

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These are good.

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Thank God I have the cut and paste feature.

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So they ask us, which inequality does the shaded

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region of the graph represent?

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So once again, it'll be good just to figure out what the

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equation of this line is.

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We could immediately say, OK, the y-intercept is 2.

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And what's its slope?

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When x increases by 1, what happens to y?

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y is going down by 3.

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So the y-intercept is equal to 2.

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We figured that out. b is equal to 2.

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The slope is equal to change in y over change in x.

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When x increases by 1, the slope-- sorry.

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The change in y is minus 3.

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It goes down by 3.

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So the slope is equal to minus 3.

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So the equation of this line is y is equal to

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minus 3x plus 2.

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mx plus b.

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And if I look at the choices, already it looks a little bit

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different than that, but we'll get to that.

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But let's see what the inequality is.

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So first of all, it's going to be either greater than or

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equal or less than or equal.

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And we say the or equal because they

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filled in the line.

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So let's think about it.

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If we pick any x, say when x is equal to 2, then according

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to this equation, according to this line, y would be

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equal to minus 4.

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Now, are we including the y's that are greater than that

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value as well or are we including the y's that are

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less than that value?

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Well, we're including the ones that are less than that point

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for a given x.

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So the shaded region is y is less than or equal to

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minus 3x plus 2.

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And you can eyeball this.

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You can kind of say, oh, this is below the graph, so it's

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less than or equal.

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And that equal was because they shaded in the line.

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If it was a dotted line it would just be less than.

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Once again, this isn't the exact format

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that they have here.

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They put all the x's and the y's on the same side, and we

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can do that.

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Let's just add 3x to both sides of this equation and you

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get 3x plus y is less than or equal to.

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When you add 3x on the right-hand side that

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just becomes 0.

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So less than or equal to 2.

262
00:11:29,690 --> 00:11:34,800
And that is 3x plus y is less than or equal to positive 2.

263
00:11:34,799 --> 00:11:38,759
And that is choice A.

264
00:11:38,759 --> 00:11:40,700
And I'm all out of time, see you in the next video.

265
00:11:40,700 --> 00:11:41,950