Hey, guys,
welcome to Algebra I.
Today's lesson
will focus on
the properties of inequality.
Back when
we were solving equations,
we spent time covering
the properties of equality.
Now we're moving on
to inequalities
and the properties
that deal with those problems.
Ready? Let's go.

(Describer) She gets a stylus and presses buttons on a screen.

Okay.
Before we jump
into inequalities,
let's think back for a bit
and make sure we're familiar
with all of the symbols
for inequality.
'Cause we know for inequality,
it means that there's a value
that's less than
or greater than something,
or greater than or equal to,
or not equal to.
There's a lot of symbols.
For this one,
the less than symbol--

(Describer) ...like the end of an arrow pointing left.

Okay.
And a trick to keeping straight
less than and greater than
is if you were to expand...
Back in first grade,
we'd call it
"the alligator mouth."
If you opened it up,
it looks like an L.
If you look at
that little alligator mouth,
that's also the left,
looks like a little L.
So, that's less than. Okay.
This symbol, when I have
the little line underneath it,
that is less than or equal to.

(Describer) She writes beside the symbol.

Okay.
Then this one,
that was greater than.

(Describer) ...like the end of an arrow pointing right.

Before we move on,
let's play around with this.
So, greater than, for example.
If you had an inequality
that said x
was greater than five...
Let's think about
some solutions to this.
Basically any number
that's greater than five
would make this inequality true.
So, six, that would work.
Seven would work.
1,200 would work.
75 million would work.
Would five work?
No. Because five is not
actually greater than five.
Would two work?
No. Because two
is not greater than five.
This is just a way of saying
x is greater than some value.
I want to jog your memory
with what inequalities meant
while we tackle reviewing
these symbols of inequality.
Let's keep moving through.
Here, greater than or equal to.

(Describer) It's a greater than symbol with the short horizontal line under it.

Okay.
And the last one,
if you run into
that inequality symbol,
that is "not equal to."

(Describer) An equals sign with a diagonal line through it.

You don't run into
this one too often,
but every now and again,
it will pop up in Algebra I.
Let's keep moving on
to the properties.
Okay, so, the transitive
property of inequality.
For all of these properties,
we are going to assume that
a, b, and c are real numbers.
Let's interpret this.
If a is less than b,
and b is less than c,
then the transitive property
says that a is less then c.
Let me tell you what I mean.
Let's say a is 1,
b is 2, and c is 3.
Let's just assign them
some values.
What this is saying is
if a is less than b--
if 1 is less than 2--
and b is less than c--
so, and 2 is less than 3--
then a is less than c.
Then is 1 less than 3.
It's a play
off of the chain rule.
If 1 is less than 2,
and 2 is less than 3,
then 1 is less than 3.
That's the transitive
property of inequality.
The same thing is true if we
are working with greater than.
If a is greater than b,
and b is greater than c,
then a is greater than c.
Same thing is true.
Let's keep going.
Addition property
of inequality. Okay.
If a is less than b,
then a plus c
will be less then b plus c.
Let me show you what I mean.
Let's just stick with a is 1,
b is 2, and c is 3
for this case.
This is saying,
"If a is less than b."
If 1 is less than 2,
then a plus c--
then 1 plus 3--
is less than b plus c--
2 plus 3.
Let's see if that's true.
1 plus 3, that's 4.
2 plus 3, that's 5. Okay.
So, that's true.
It's saying that
if you have two values,
and one is smaller
than the other,
if you add the same quantity
to each value,
that same relationship
is still true.
Whatever number was smaller
before you
added something to it
is still gonna be smaller
after you add something
to both sides.
That's all the addition property
of inequality is saying.
Same thing is true
for greater than.
If a is greater than b,
then a plus c
is greater than b plus c.
All right, let's keep going.
Subtraction property
of inequality. All right.
If a is less than b--
you probably know
what I'm about to write down--
then a minus c...
is less than b minus c.
Okay?
Let's play around
with some numbers here.
I was gonna try
to avoid negative numbers
and not make it too complicated,
but I think you can handle it.
So, a is 1,
b is 2, and c is 3.
So, a is less than b--
1 is less than 2--
then a minus c--
so, 1 minus 3--
is less than b minus c--
so, less than 2 minus 3.
All right.
Let's see if that's true.
1 minus 3, that's -2.
2 minus 3, that's -1.
And -2 is smaller than -1.
That's what I wasn't sure
if you remembered,
but I'll jog your memory
just in case.
Think back to the number line.
Let's throw up
a quick number line.

(Describer) She draws a line with arrows at the ends.

We don't need much on it.
Let's just get
0, 1, 2, 3, -1, -2.
So, remember--
Let me clean that up so you're
clear that that is a 2 there.
It's easy to tell values
that are larger or smaller
when we're on the positive side
of the number line.
Intuitively, we know
that 3 is bigger than 2.
Ten--that would be
way over here--
is bigger than 1.
We know that
on the positive side.
Remember the negative side?
The further away
you get from 0,
the smaller
your value actually is.
In this case,
-2 is smaller than -1
because it's further
away from 0
on the negative side
of my number line.
I know that's tricky,
because you're like,
"But 2 is bigger than 1."
But -2 is smaller than -1.
Because on the negative side,
the further away you are from 0,
the smaller
your value actually is.
So, this subtraction
of inequality holds true.
All right, let's keep going.
Same thing's true
for greater than.
So, if a is greater than b,
a minus c will also be
greater than b minus c.
Okay.
Multiplication.
Things start getting
interesting here.
Not at first, but they will.
If a is less than b
and c is positive,
then a times c
will be less then b times c.
Let me show you what I mean.
Let's say a is 1,
b is 2, c is 3.
They've been doing good
so far.
So, a is less than b,
1 is less than 2.
C is positive.
Yes, c is a positive number.
So, a times c, 1 times 3,
will be less then b times c,
or 2 times 3, okay?
So, 1 times 3, that's 3,
2 times 3, that's 6.
Okay, that's true.
If a is less than b
and c is positive,
the product of a and c
will be less than
the product of b and c.
Let's keep going.
The same thing holds true
for greater than.
If a is greater than b
and c is positive,
the product of a and c
will be greater than
the product of b and c.
Here's where it
gets interesting.
You may remember this
from pre-algebra also.
Things change if c is negative.
If a is less then b
and c is negative,
then ac is
actually greater than bc.
Let me show you
what I mean.
I'm gonna switch
the values around.
I'm gonna let a be 3
and b be 2,
and I'm gonna let C be -1
to keep things simple.
Oop, that's not gonna work,
'cause a's not smaller than b.
So, let's let...
a be 2 and c be 3.
There we go.
So, a is smaller than b,
a is less than b. It is.
Two is less than 3,
and c is negative.
It is, because
we're letting c be -1.
I'm gonna not write
the inequality sign
till the very end
to show you what I mean.
The product of a and c,
that would be 2 times -1.
That's -2.
The product of b and c,
that would be 3 times -1.
That's -3.
You see that
the product of a and c
is greater than
the product of b and c.
That always happens
if your c is negative.
If you multiply
an inequality,
if you multiply both sides
by a negative number,
essentially your inequality sign
flips the other way.
Here we had a less than b
and c was negative.
So, the product of a and c
is greater than
the product of b and c.
As a shortcut way
to think about it,
if you're multiplying both sides
by a negative number,
you flip your inequality sign
for your answer.
Okay? Keep going.
Same thing's true
for greater than.
If a is greater than b
and c is negative,
then the product of a and c
will be less than
the product of b and c.
We started out
with greater than,
but in our answer,
inequality switched to less than
because we were multiplying
by a negative number.
So, division property
of inequality.
Same idea for multiplication.
Okay. If c is positive,
things are kind of normal.
If a is less than b
and c is positive,
then a divided by c
will be less than
b divided by c.
I think you're getting the idea
of how that works.
I won't bother doing
a numeric example for that one.
Same thing's true
for greater than.
If a is greater than b
and c is positive,
the quotient, a divided by c,
will be greater than
b divided by c.
Same thing happens here
like it did with
the multiplication property.
If a is less then b
and c is negative,
that inequality sign
will switch the other way.
So, a divided by c
will be greater than
b divided by c.
And that happens
if that c is negative.
Same idea
for the greater than case.
If a is greater than b
and c is negative,
then a divided by c will be
less than b divided by c.
Let's look at an example
and see if you can spot
these properties
while you're solving
an inequality.
This will throw you back
to pre-algebra,
when you were solving
multi-step linear inequalities.
We're gonna zone in
where we see property symbols,
or property boxes,
and figure out
what property allowed us
to move between those steps
when we get down there.
Let's interpret this.
Step one, just giving us
the problem there.
Five times that quantity
x plus 2 is greater than 30.
Step two looks like they applied
the distributive property,
right?
'Cause now we have
5x plus 10 is greater than 30.
Step three,
looks like they are
subtracting 10 from both sides.
Right?
They've got 5x plus 10 minus 10
is greater than 30 minus 10.
In step four,
5x is greater than 20.
Okay. And we see
the little property question.
Which property of inequality
tells us that if we subtract
the same quantity
from both sides
of our inequality,
that inequality
will still hold true--
whatever
that relationship was?
Remember?
It was that subtraction
property of inequality.
Okay? All right,
let's keep moving.
From step four to step five,
let's scroll a bit.
We divided both sides by 5,
and from step five to step six,
we went from 5x divided by 5
is greater than 20 divided by 5
to x is greater than 4.
We divided both sides by 5,
and the relationship between
both sides of our inequality
still held true.
Which property of inequality
told us that--
that if we divide both sides
by a positive number,
the relationship
still holds true,
whatever it was originally?
That was the division property
of inequality. Okay?
All right. Good job on that.
Take a look at this one.
I'm gonna give you a bit.
I want you to focus on
steps one through four.
Do you know what property
allowed us to move between
step three and step four?
Then I'll scroll down,
because there's
one more property question
lower in this problem.
Hit pause,
give that some thought,
and press play when you're
ready to go over it.

(Describer) Titles:
Step 1: Negative x minus 4 is less than negative 2 times x plus 3.
Step 2: Negative x minus 4 is less than negative 2x minus 6.
Step 3: Negative x minus 4 plus 2x is less than negative 2x plus 2x minus 6. By this step is box with the word "property" with a question mark.
Step 4: x plus 4 is less than negative 6.

(female describer)
Step 1: -x minus 4
is less than -2 times x plus 3.
Step 2: -x minus 4
is less than -2x minus 6.
Step 3: -x minus 4 plus 2x
is less than -2x
plus 2x minus 6.
By this step is the box
with the word "Property"
with a question mark.
Step 4: x plus 4
is less than -6.
As you made
your way through here
and got to step three
and step four,
and you saw that you had
2x added to both sides,
and that initial relationship
between the left side
and right side of our inequality
still held true.
The addition property
of inequality
allowed that to happen.
Good job if you got that.
If not, review those properties
and you'll get good at it.
I'm gonna scroll down.
Now focus on step four
through step six
and see if you can figure out
what property allowed you
to move between
steps five and six.
Press pause,
give that some thought,
then press play
when you're ready to check.

(Describer) Titles:
Step 5: x plus 4 minus 4 is less than negative 6 minus 4. By this step is a box with the word "property" and a question mark.
Step 6: x is less than negative 10.

(describer)
Step 5:
x plus 4 minus 4
is less than -6 minus 4.
By this step is a box
with the word "Property"
and a question mark.
Step 6:
x is less than -10.
Hope you gave that some thought.
If you looked at
what was going on
between steps five and six,
you subtracted 4 from both sides
and the initial inequality
relationship still held true:
x was less than -10.
That was the subtraction
property of inequality
that allowed you to move
between those two steps.
Great job
getting familiar with
the properties of inequality.
I hope to see you back soon
for more Algebra I. Bye.

(Describer) Accessibility provided by the U.S. Department of Education.