WEBVTT
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Hi, and welcome.
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This is Anthony
Varela and today we're
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going to solve absolute
value equations.
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So we're going to start
with a simple example
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where we just have the absolute
value of x equals some value.
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Then we're going to
see a linear expression
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inside absolute
value bars and talk
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about how to solve those
types of equations.
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And then our last example,
will have an expression
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inside of an absolute value
bar and then something outside
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of it too on the same
side of the equals sign.
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So first, let's
review absolute value.
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Now, recall that the
absolute value of a number
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is its distance away from
0 on the number line.
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So if we put 4 on
the number line,
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you would see that
4 is four units away
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from 0 on the number line.
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And on the other side of 0,
we can see that negative 4
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is also four units away
from 0 on the number line.
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So we could say then that the
absolute value of 4 equals 4,
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and the absolute value of
negative 4 equals 4 as well.
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Now I'd like to pay
particular attention
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to this absolute
value of negative 4.
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So if we think about
negative 4 being
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inside of the
absolute value bar,
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we had to take the
opposite of that in order
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to express it as positive 4.
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So the opposite of
negative 4 is positive 4,
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and that's the absolute value.
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So we get into an
interesting definition
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then of absolute value.
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And this is what we call
the piecewise definition
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to absolute value.
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And it's OK if you don't quite
understand piecewise functions.
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But essentially, this
just means that we're
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evaluating this expression,
the absolute value of x,
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differently depending
on what x is.
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And now when x is greater
than or equal to 0,
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we don't really do anything.
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We can erase the absolute
value bars and we'd be fine.
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So the absolute value
of x is x when x
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is greater than or equal to 0.
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Now, when x is less than
0, so when x is negative,
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then we evaluate it as
the opposite of that.
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So that would be
this example right.
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Here we had a negative value
inside the absolute value bars.
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So we took the opposite of
that, which would be positive 4.
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So we're going to be using
this piecewise definition
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of absolute value to solve
some absolute value equations.
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So we'll start off with a simple
example, the absolute value
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of x equals 5.
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Now in order to
solve this equation,
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we're going to make two
different equations out
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of this from our
piecewise definition here.
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So one of the equations
is going to be x equals 5.
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And the other is going to
be negative x equals 5.
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So when we took away
the absolute value bars,
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we were able to just keep it as
it is minus the absolute value
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bars.
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And then we also
the second equation
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where we're taking the
opposite of this expression, so
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negative x equals 5.
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Well, we already have then one
of our solutions, x equals 5.
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If negative x equals 5,
then x equals negative 5.
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So these are two solutions
then to this absolute value
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equation.
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x could be 5, x could
also be negative 5.
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When you take the
absolute value, we get 5.
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So putting this on a graph--
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So here's a graph of
the absolute value of x.
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And then we're going to
have y equals 5 here.
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So the intersections
then to y equals
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the absolute value
of x and y equals
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5 are at negative
5 and positive 5.
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So now let's get a little
bit more complicated.
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Here we have a linear expression
inside of our absolute value
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bars.
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So how can we solve the absolute
value of 2x minus 3 equals 3?
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Well, we're going to
create, once again,
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two different equations.
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Our first equation is
going to be 2x minus 3,
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we're just erasing the
absolute value bars.
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But our other
equation we're going
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to take the negative of
the entire expression.
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So it's very important
that I'm grouping
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my mx plus b in
parentheses and putting
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a negative out in front.
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So let's go ahead and work with
our first equation, 2x minus 3
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equals 3.
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Well, I'll add 3 to both
sides of the equation
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and then divide by 2.
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So one of my solutions
is x equals 3.
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Well, my other
equation is going to be
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negative 2x minus 3 equals 3.
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And so first what
I'm going to do
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is distribute that negative
throughout 2x minus 3.
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So we have negative
2x plus 3 equals 3.
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And to solve for x, I am going
to subtract 3 from both sides.
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So if negative 2x equals
0, then x just equals 0.
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So I have my two solutions,
x equals 3 and x equals 0.
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Let's go ahead and take a
look at this on the graph.
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So this is y equals the
absolute value of 2x minus 3.
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And now I'm also going
to plot y equals 3
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and look at our points
of intersection.
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Those occur at x
equals 0 and x equals
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3, our solutions for x here.
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One final example, here we have
an absolute value expressions
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here, so we have the absolute
value of negative x plus 2.
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And then we have something
outside of the absolute value
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bars, but it's still on the
side of the equals sign that
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contains the absolute value.
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So how do we deal with
something like this?
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Well, we'd like to
remove it from that side.
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So we want to have only the
absolute value expression
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on that side of the equation.
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So I'm just going to
subtract 1 from both sides
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of the equation.
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And so I'm going to move
it to the other side.
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So really, I am dealing
with an equivalent equation,
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the absolute value of
negative x plus 2 equals 3.
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And this fits this form.
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So it's just like
the previous example.
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So one of our equations is
going to be negative x plus 2
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equals 3.
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And then I'm going to
group negative x plus 2
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and apply a negative,
so taking the opposite
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of this expression.
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That also equals 3.
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So solving for negative
x plus 2 equals 3,
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I'll subtract 2 from both sides.
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And if negative x equals 1,
then x equals negative 1.
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That's one of our solutions.
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For this equation here,
first what I'll do
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is distribute that negative.
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So I have a positive
x minus 2 equals 3.
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And then I'll add 2 to
both sides of the equation.
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And the other solution
is x equals 5.
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Let's once again confirm
this on the graph.
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So this is the graph of y
equals the absolute value
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of negative x plus 2 plus 1.
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And then I have y equals 4.
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And our points of intersection
occur at negative 1 and 5.
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Those were our solutions to x
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So let's review our notes
on solving absolute value
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equations.
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We talked about this piecewise
definition of absolute value.
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When x is greater
than or equal to 0,
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we can just evaluate the
absolute value of x as x.
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If x is a negative, so
that would be less than 0,
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then we have to take
the opposite of what we
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see in the absolute value bars.
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So when solving absolute
value equations,
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you're taking the
absolute value equation
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and you're really making
two separate equations.
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And we saw that in
all of our examples.
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One equation you're just are
racing the absolute value bars.
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But in your other
equation, your grouping
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in that entire expression and
taking the opposite of that,
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or multiplying it by negative 1.
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So thanks for watching this
tutorial on absolute value
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equations.
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Hope to see you next time.