Absolute Value Inequalities 3
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>> Julie Harland: Hi,
this is Julie Harland
and I'm your math gal.
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This is Part III
of Absolute Value Inequalities
and we saw these two
inequalities on this video.
Alright, we're going
to solve this absolute
value inequality.
Absolute value
of 2 X minus 3 plus 2 is less
than or equal to 5.
Alright, so remember
if we isolate the absolute
value sign, then we have a way
of rewriting the inequality
without absolute values.
So that's the first thing I
want to do here.
This plus 2 is not inside the
absolute value so I am going
to subtract 2 from both sides
so that I could isolate the
absolute value.
So that gives me the absolute
value of 2 X minus 3 is less
than or equal to 3.
Now we have the absolute value
of something less
than some positive number -
right, less than or equal
in this case -
to some positive number.
So remember what we do.
We can take the part
underneath the absolute value
sign: 2 X minus 3 and we know
that can go
between negative 3 and 3.
Remember this is
because absolute value has
to do with distance
and we want some distance
to zero to be less
than 3 spaces away.
Okay, so now we need
to solve this.
So we can add 3 to both sides.
Do it again.
So I have zero is less than
or equal to 2 X is less than
or equal to 6.
And lastly,
we want to divide everything
by 2 so we can solve for X.
So remember zero
over 2 is zero.
So our answer is that X is
in between zero and 3.
So how would we write
that in interval notation?
It's in between zero and 3.
If we were going
to graph the solution,
so let's say this is zero
and 3 it would be
in between zero and 3.
Now what we could do is do a
little check you know.
Like click --
let's pick a number
in between zero and 3 like 1
or 2, right?
Those are easy numbers
to check and plug it
in this original inequality
and see if it's a solution.
Now the original remember had
the absolute value
of 2 X minus 3 plus 2 is less
than or equal to 5.
So I'm going just make this a
little bit smaller
so we have some room
to do the check right here
on the side.
In fact I'm going
to take away the word
"solve" just
to give ourselves even a
little bit more space.
And let's see.
So how about if we just check?
How about putting in 2 for X
and see if it works?
So here's our original
problem: 2 X minus 3 is
in the absolute value plus 2
is less than or equal to 5.
And we're just going to plug
in 2 for X. And the question,
is this going
to be a true statement?
So inside the absolute value
sign we've got 4 minus 3,
right?
Because we have
to do 2 times 2.
Four minus 3 is 1.
And then what's the absolute
value of 1?
Well that's 1.
So I've got the 1 plus 2
which is 3 is less than
or equal to 5.
Is that true?
Is 3 either less than 5
or equal to 5?
Yes because it's less than 5.
So we know
that 2 is a solution.
So it looks
like at least a number
between zero and 3 is correct.
And the only way to tell
if these are the right end
points is by plugging in zero.
And when you plug in zero,
you should get the --
those sides are actually equal
to each other.
And when you plug in 3
at the other end point,
you should get the both sides
end of it being equal
to each other.
So it looks
like this is our correct
solution and interval notation
zero 3.
Here's a problem for you
to solve so try this
on your own
and you will have some
fractions involved
in your answer.
That's okay.
And just write your answer
in interval notation.
Alright so let's see.
How do we solve this?
We don't have the absolute
value sign isolated so how
about we add 2 to both sides?
So if we do that I've got
on the left side just absolute
value of 3 X plus 1.
And on the right side I've
got 5.
Alright, so now the number --
I'm sorry.
The part inside the absolute
value sign is 3 X plus 1.
When you have less than
or less than or equal,
remember it goes
in between the negative
and positive of this number
on the right
when it's positive.
So that's in between negative
5 and positive 5.
So now we just solve this
compound inequality.
Let's subtract 1
from both sides to get started
so that we could isolate the 3
X. So that gives us negative 6
is less than or equal
to 3 X is less than
or equal to 4.
And then we want
to divide everything by 3
to solve for X.
So we get negative 2 is less
than or equal to X is less
than four-thirds which is one
and a third.
You could write four-thirds
or one and a third.
This looks
like it's our solution.
So an interval notation --
oops, I forgot the less
than or equal.
In interval notation,
that's in between negative 2
and four-thirds or one
and one third.
Remember we used the brackets
when you've got the
equal sign.
Use parentheses
if it's just strictly less
than or strictly greater than.
Alright now,
you can check some number
in between negative 1 --
negative 2 and one
and one-third like zero.
Right? That might be an easy
one to check.
So let's just check
to make sure X equals zero
really is a solution.
Now remember
when I do these checks?
This doesn't mean I've got the
correct answer
because I can't check every
single number.
I'm just checking
if one number
in between there makes the
original inequality true.
So I'm just going to work
on the left hand side here.
I have 3 times zero is what
I'm plugging in for X.
And then inside the absolute
value, 3 times zero is zero
so I have zero plus 1 - okay -
which is 1.
And now I've got the absolute
value of 1 which is 1.
Right? So I end
up with 1 minus 2.
And then 1 minus 2 is
negative 1.
So the question is,
when I plug in zero,
will I get a number
on the left side that's less
than or equal to 3?
That's basically what I'm
checking out here.
And yes, that is
absolutely true.
So at every single point,
in the end I'm asking myself,
"Less than or equal to 3?"
Yes, that is true.
So it looks
like at least I've got the
right range.
And again, the only way
to know whether your end
points negative 2 or one
and one third are correct is
to actually plug in negative 2
for X, right?
And also plug in 1
and one third for X. So just
because a lot
of people have trouble
with fractions,
I'm going to go ahead and plug
in the one and one third
and we'll try that.
So let's go ahead and plug
in four-thirds for X.
So here's our original.
We have 3 times four-thirds
plus 1 minus 2 and we're going
to see if that's going
to be less than or equal to 3.
So we have
to simplify inside the
absolute value.
Threes cancel.
So I just get 4 plus 1 inside
the absolute value.
And 4 plus 1 is 5
so I just get 5
in the absolute value.
And now what is the absolute
value of 5?
That will be 5.
So if 5 minus 2 --
and that's 3.
And the question is this:
"Is that less than
or equal to 3?"
Yes, because 3 is equal to 3.
Remember when I write 3 is
less than or equal to 3?
I'm saying, "Is this number
on the left either less
than the number on the right
or is it equal to the number
on the right?
As long as one
of those things is true,
it works.
So four-thirds or one
and one third was a solution.
Alright, so we've just done a
few more problems
with the less than or equal.
And if you go
onto the next video,
we're going
to do the tricky problems.
Tricky problems with less than
or less than
or equal inequalities
with the absolute value
and then after that we will go
on to the greater than
or equal problems or great
than with the absolute value.
Please visit my website
at www.yourmathgal.com
where you can view all
of my videos
which are organized by topic.
