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Let's do some equations that deal with absolute values.
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And just as a bit of a review,
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when you take the absolute value of a number.
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Let's say I take the absolute value of -1.
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What you're really doing is
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you're saying, how far is that number from 0?
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And in the case of -1, if we draw a number line right there
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-- that's a very badly
drawn number line.
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If we draw a number line right there, that's 0.
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You have a -1 right there.
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Well, it's 1 away from 0.
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So the absolute value of -1 is 1.
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And the absolute value of 1 is also 1 away from 0.
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It's also equal to 1.
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So on some level, absolute value is the distance from 0.
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But another, I guess simpler way to think of it,
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it always results in the positive version of the number.
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The absolute value of -7,346 is equal to 7,346.
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So with that in mind, let's try to
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solve some equations with absolute values in them.
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So let's say I have the equation
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the absolute value of x -5 is equal to 10.
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And one way you can interpret this,
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and I want you to think about this, this is actually saying
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that the distance between x and 5 is equal to 10.
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So how many numbers that are exactly 10 away from 5?
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And you can already think of the solution to this equation,
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but I'll show you how to solve it systematically.
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Now this is going to be true in two situations.
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Either x -5 is equal to +10.
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If this evaluates out to +10,
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then when you take the absolute value of it,
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you're going to get +10.
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Or x - 5 might evaluate to -10.
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If x - 5 evaluated to -10, when you take the absolute value of it,
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you would get 10 again.
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So x - 5 could also be equal to -10.
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Both of these would satisfy this equation.
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Now, to solve this one,
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add 5 to both sides of this equation.
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You get x is equal to 15.
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To solve this one, add 5 to both sides of this equation.
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x is equal to -5.
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So our solution,
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there's two x's that satisfy this equation.
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x could be 15.
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15 - 5 is 10, take the absolute value,
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you're going to get 10, or x could be -5.
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- 5 minus 5 is -10.
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Take the absolute value, you get 10.
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And notice, both of these numbers
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are exactly 10 away from the number 5.
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Let's do another one of these.
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Let's do another one.
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Let's say we have
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the absolute value of x + 2 is equal to 6.
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So what does that tell us?
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That tells us that either x + 2,
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that the thing inside the absolute value sign, is equal to 6.
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Or the thing inside of the absolute value sign,
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the x + 2, could also be -6.
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If this whole thing evaluated to -6,
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you take the absolute value, you'd get 6.
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So, or x + 2 could equal -6.
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And then if you subtract 2 from both sides of this equation,
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you get x could be equal to 4.
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If you subtract 2 from both sides of this equation,
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you get x could be equal to -8.
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So these are the two solutions to the equation.
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And just to kind of have it gel in your mind,
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that absolute value, you can kind of view it as a distance,
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you could rewrite this problem
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as the absolute value of x minus -2 is equal to 6.
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And so this is asking me,
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what are the x's that are exactly 6 away from -2?
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Remember, up here we said,
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what are the x's that are exactly 10 away from +5?
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Whatever number you're subtracting from +5,
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these are both 10 away from +5.
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This is asking,
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what is exactly 6 away from -2?
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And it's going to be 4, or -8.
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You could try those numbers out for yourself.
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Let's do another one of these.
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Let's do another one, and we'll do it in purple.
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Let's say we have the absolute value of 4x.
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I'm going to change this problem up a little bit.
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4x -1.
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The absolute value of 4x -1, is equal to--
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actually, I'll just keep it-- is equal to 19.
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So, just like the last few problems,
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4x -1 could be equal to 19.
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Or 4x -1 might evaluate to -19.
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Because then when you take the absolute value,
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you're going to get 19 again.
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Or 4x -1 could be equal to -19.
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Then you just solve these two equations.
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Add 1 to both sides of this equation--
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we could do them simultaneous, even.
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Add 1 to both sides of this, you get 4x is equal to 20.
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Add 1 to both sides of this equation,
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you get 4x is equal to -18.
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Divide both sides of this by 4, you get x is equal to 5.
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Divide both sides of this by 4, you get x is equal to -18/4,
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which is equal to -9/2.
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So both of these x values satisfy the equation.
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Try it out.
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-9/2 x 4.
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This will become a -18.
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-18 minus 1 is -19.
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Take the absolute value, you get 19.
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You put a 5 here, 4 x 5 is 20.
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Minus 1 is +19.
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So you take the absolute value.
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Once again, you'll get a 19.
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Let's try to graph one of these, just for fun.
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So let's say
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I have y is equal to the absolute value of x +3.
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So this is a function, or a graph,
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with an absolute value in it.
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So let's think about two scenarios.
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There's one scenario
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where the thing inside of the absolute value is positive.
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So you have the scenario where x + 3
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I'll write it over here: x + 3 is > 0.
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And then you have the scenario where x +3 is < 0.
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When x +3 is > 0,
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this graph, or this line--or I guess we can't call it a line--
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this function, is the same thing as y is equal to x +3.
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If this thing over here is > 0,
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then the absolute value sign is irrelevant.
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So then this thing is the same thing
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as y is equal to x +3.
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But when is x +3 > 0?
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Well, if you subtract 3 from both sides,
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you get x is > -3.
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So when x is > -3,
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this graph is going to look just like y is equal to x +3.
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Now, when x +3 is < 0.
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When the situation where this--
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the inside of our absolute value sign--is negative,
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in that situation this equation is going to be
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y is equal to the negative of x +3.
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How can I say that?
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Well, look, if this is going to
be a negative number, if x
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plus 3 is going to be a negative
number-- that's what
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we're assuming here-- if it's
going to be a negative number,
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then when you take the absolute
value of a negative
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number, you're going to
make it positive.
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That's just like multiplying
it by negative 1.
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If you know you're taking the
absolute value of a negative
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number, it's just like
multiplying it by negative 1,
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because you're going to
make it positive.
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And this is going to
be the situation.
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x plus 3 is less than 0.
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If we subtract 3 from both
sides, when x is less than
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negative 3.
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So when x is less than negative
3, the graph will
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look like this.
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When x is greater than negative
3, the graph will
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look like that.
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So let's see what that
would make the
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entire graph look like.
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Let me draw my axes.
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That's my x-axis, that's
my y-axis.
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So just let me multiply this
out, just so we have it in mx
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plus b form.
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So this is equal to negative
x minus 3.
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So let's just figure out
what this graph would
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look like in general.
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Negative x minus 3.
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The y-intercept is negative
3, so 1, 2, 3.
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And negative x means it
slopes downward, has a
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downward slope of 1.
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So it would look like this.
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The x-intercept would be
at x is equal to--.
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So if you say y is equal to 0,
that would happen when x is
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equal to negative 3.
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So it's going to go
through that line,
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that point right there.
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And the graph, if we didn't
have this constraint right
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here, would look something
like this.
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That's if we didn't constrain
it to a certain interval on
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the x-axis.
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Now this graph, what
does it look like?
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Let's see.
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It has its y-intercept
at positive 3.
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Just like that.
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And where's its x-intecept?
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When y is equal to 0,
x is negative 3.
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So it also goes through that
point right there, and it has
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a slope of 1.
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So it would look something
like this.
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That's what this graph
looks like.
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Now, what we figured out is
that this absolute value
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function, it looks like this
purple graph when x is less
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than negative 3.
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So when x is less than negative
3-- that's x is equal
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to negative 3 right there-- when
x is less than negative
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3, it looks like this
purple graph.
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Right there.
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So that's when x is less
than negative 3.
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But when x is greater than
negative 3, it looks like the
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green graph.
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It looks like that.
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So this graph looks like
this strange v.
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When x is greater than negative
3, this is positive.
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So we have the graph of-- we
have a positive slope.
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But then when x is less than
negative 3, we're essentially
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taking the negative of the
function, if you want to view
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it that way, and so we have
this negative slope.
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So you kind of have this
v-shaped function, this
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v-shaped graph, which is
indicative of an absolute
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value function.
