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Absolute value inequalities | Linear equations | Algebra I | Khan Academy - YouTube
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I now want to solve some
inequalities that also have
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absolute values in them.
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And if there's any topic in
algebra that probably confuses
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people the most, it's this.
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But if we kind of keep our head
on straight about what
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absolute value really means, I
think you will find that it's
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not that bad.
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So let's start with a nice,
fairly simple warm-up problem.
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Let's start with the absolute
value of x is less than 12.
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So remember what I told
you about the
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meaning of absolute value.
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It means how far away
you are from 0.
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So one way to say this is, what
are all of the x's that
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are less than 12 away from 0?
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Let's draw a number line.
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So if we have 0 here, and we
want all the numbers that are
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less than 12 away from 0, well,
you could go all the way
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to positive 12, and you could go
all the way to negative 12.
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Anything that's in between these
two numbers is going to
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have an absolute value
of less than 12.
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It's going to be less
than 12 away from 0.
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So this, you could say, this
could be all of the numbers
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where x is greater
than negative 12.
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Those are definitely going to
have an absolute value less
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than 12, as long as they're
also-- and, x has to
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be less than 12.
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So if an x meets both of these
constraints, its absolute
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value is definitely going
to be less than 12.
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You know, you take the absolute
value of negative 6,
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that's only 6 away from 0.
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The absolute value of negative
11, only 11 away from 0.
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So something that meets both
of these constraints will
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satisfy the equation.
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And actually, we've solved it,
because this is only a
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one-step equation there.
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But I think it lays a good
foundation for the next few
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problems. And I could actually
write it like this.
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In interval notation, it would
be everything between negative
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12 and positive 12, and not
including those numbers.
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Or we could write it like this,
x is less than 12, and
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is greater than negative 12.
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That's the solution
set right there.
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Now let's do one that's a little
bit more complicated,
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that allows us to think
a little bit harder.
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So let's say we have the
absolute value of 7x is
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greater than or equal to 21.
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So let's not even think about
what's inside of the absolute
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value sign right now.
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In order for the absolute
value of anything to be
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greater than or equal to
21, what does it mean?
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It means that whatever's inside
of this absolute value
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sign, whatever that is inside of
our absolute value sign, it
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must be 21 or more
away from 0.
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Let's draw our number line.
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And you really should visualize
a number line when
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you do this, and you'll never
get confused then.
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You shouldn't be memorizing
any rules.
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So let's draw 0 here.
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Let's do positive 21, and let's
do a negative 21 here.
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So we want all of the numbers,
so whatever this thing is,
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that are greater than
or equal to 21.
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They're more than
21 away from 0.
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Their absolute value
is more than 21.
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Well, all of these negative
numbers that are less than
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negative 21, when you take their
absolute value, when you
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get rid of the negative sign,
or when you find their
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distance from 0, they're all
going to be greater than 21.
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If you take the absolute value
of negative 30, it's going to
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be greater than 21.
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Likewise, up here, anything
greater than positive 21 will
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also have an absolute value
greater than 21.
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So what we could say is 7x needs
to be equal to one of
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these numbers, or 7x needs to
be equal to one of these
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numbers out here.
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So we could write 7x needs to
be one of these numbers.
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Well, what are these numbers?
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These are all of the numbers
that are less than or equal to
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negative 21, or 7x-- let me do a
different color here-- or 7x
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has to be one of
these numbers.
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And that means that 7x has to
be greater than or equal to
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positive 21.
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I really want you to
kind of internalize
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what's going on here.
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If our absolute value is greater
than or equal to 21,
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that means that what's inside
the absolute value has to be
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either just straight up greater
than the positive 21,
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or less than negative 21.
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Because if it's less than
negative 21, when you take its
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absolute value, it's going to
be more than 21 away from 0.
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Hopefully that make sense.
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We'll do several of these
practice problems, so it
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really gets ingrained
in your brain.
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But once you have this set up,
and this just becomes a
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compound inequality, divide both
sides of this equation by
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7, you get x is less than
or equal to negative 3.
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Or you divide both sides of
this by 7, you get x is
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greater than or equal to 3.
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So I want to be very clear.
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This, what I drew here, was
not the solution set.
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This is what 7x had
to be equal to.
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I just wanted you to visualize
what it means to have the
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absolute value be greater
than 21, to be more than
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21 away from 0.
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This is the solution set. x
has to be greater than or
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equal to 3, or less than
or equal to negative 3.
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So the actual solution set to
this equation-- let me draw a
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number line-- let's say that's
0, that's 3, that is negative
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3. x has to be either greater
than or equal to 3.
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That's the equal sign.
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Or less than or equal
to negative 3.
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And we're done.
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Let's do a couple
more of these.
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Because they are, I think,
confusing, but if you really
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start to get the gist of what
absolute value is saying, they
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become, I think, intuitive.
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So let's say that we have
the absolute value-- let
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me get a good one.
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Let's say the absolute value of
5x plus 3 is less than 7.
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So that's telling us that
whatever's inside of our
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absolute value sign has to be
less than 7 away from 0.
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So the ways that we can be less
than 7 away from 0-- let
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me draw a number line-- so the
ways that you can be less than
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7 away from 0, you could be less
than 7, and greater than
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negative 7.
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Right?
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You have to be in this range.
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So in order to satisfy this
thing in this absolute value
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sign, it has to be-- so the
thing in the absolute value
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sign, which is 5x plus 3--
it has to be greater than
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negative 7 and it has to be less
than 7, in order for its
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absolute value to
be less than 7.
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If this thing, this 5x plus 3,
evaluates anywhere over here,
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its absolute value, its
distance from 0, will
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be less than 7.
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And then we can just
solve these.
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You subtract 3 from
both sides.
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5x is greater than
negative 10.
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Divide both sides by 5. x is
greater than negative 2.
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Now over here, subtract
3 from both sides.
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5x is less than 4.
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Divide both sides by 5, you
get x is less than 4/5.
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And then we can draw
the solution set.
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We have to be greater than
negative 2, not greater than
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or equal to, and
less than 4/5.
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So this might look like a
coordinate, but this is also
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interval notation, if we're
saying all of the x's between
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negative 2 and 4/5.
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Or you could write it all of the
x's that are greater than
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negative 2 and less than 4/5.
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These are the x's that satisfy
this equation.
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And I really want you
to internalize this
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visualization here.
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Now, you might already be seeing
a bit of a rule here.
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And I don't want you to just
memorize it, but I'll give it
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to you just in case
you want it.
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If you have something like f of
x, the absolute value of f
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of x is less than, let's
say, some number a.
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Right?
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So this was the situation.
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We have some f of
x less than a.
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That means that the absolute
value of f of x, or f of x has
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to be less than a away from 0.
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So that means that f of x has to
be less than positive a or
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greater than negative a.
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That translates to that, which
translates to f of x greater
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than negative a and f
of x less than a.
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But it comes from
the same logic.
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This has to evaluate to
something that is less than a
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away from 0.
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Now, if we go to the other side,
if you have something of
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the form f of x is
greater than a.
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That means that this thing has
to evaluate to something that
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is further than a away from 0.
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So that means that f of x is
either just straight up
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greater than positive a, or f of
x is less than negative a.
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Right?
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If it's less than negative a,
maybe it's negative a minus
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another 1, or negative
5 plus negative a.
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Then, when you take its
absolute value, it'll
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become a plus 5.
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So its absolute value is going
to be greater than a.
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So I just want to-- you could
memorize this if you want, but
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I really want you to think about
this is just saying, OK,
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this has to evaluate, be less
than a away from 0, this has
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to be more than a away from 0.
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Let's do one more, because
I know this can be
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a little bit confusing.
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And I encourage you to watch
this video over and over and
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over again, if it helps.
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Let's say we have the absolute
value of 2x-- let me do
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another one over here.
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Let's do a harder one.
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Let's say the absolute value
of 2x over 7 plus 9 is
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greater than 5/7.
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So this thing has to evaluate to
something that's more than
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5/7 away from 0.
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So this thing, 2x over 7 plus
9, it could just be straight
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up greater than 5/7.
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Or it could be less than
negative 5/7, because if it's
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less than negative 5/7, its
absolute value is going to be
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greater than 5/7.
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Or 2x over 7 plus 9 will be
less than negative 5/7.
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We're doing this case
right here.
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And then we just solve both
of these equations.
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See if we subtract-- let's just
multiply everything by 7,
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just to get these denominators
out of the way.
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So if you multiply both sides by
7, you get 2x plus 9 times
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7 is 63, is greater than 5.
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Let's do it over here, too.
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You'll get 2x plus 63 is
less than negative 5.
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Let's subtract 63 from both
sides of this equation, and
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you get 2x-- let's see.
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5 minus 63 is 58, 2x
is greater than 58.
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If you subtract 63 from both
sides of this equation, you
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get 2x is less than
negative 68.
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Oh, I just realized I
made a mistake here.
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You subtract 63 from both sides
of this, 5 minus 63 is
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negative 58.
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I don't want to make a careless
mistake there.
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And then divide both
sides by 2.
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You get, in this case, x is
greater than-- you don't have
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to swap the inequality, because
we're dividing by a
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positive number-- negative 58
over 2 is negative 29, or,
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here, if you divide both sides
by 2, or, x is less than
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negative 34.
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68 divided by 2 is 34.
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And so, on the number line,
the solution set to that
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equation will look like this.
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That's my number line.
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I have negative 29.
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I have negative 34.
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So the solution is, I can either
be greater than 29, not
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greater than or equal to, so
greater than 29, that is that
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right there, or I could be
less than negative 34.
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So any of those are going to
satisfy this absolute value
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inequality.
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