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Absolute Value Equations

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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.
Title:
Absolute Value Equations
Description:

Absolute Value Equations

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Video Language:
English
Duration:
10:41

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