Welcome to the presentation on solving inequalities,
Welcome to the presentation on solving inequalities,

or I guess you call them algebra inequalities.
or I guess you call them algebra inequalities.

So let's get started.
So let's get started.

If I were to tell you that, well, let's just say x is
If I were to tell you that, well, let's just say x is

greater than 5, right?
greater than 5, right?

So x could be 5.01, it could be 5.5, it could be a million.
So x could be 5.01, it could be 5.5, it could be a million.

It just can't be 4, or 3, or 0, or negative 8, and actually,
It just can't be 4, or 3, or 0, or negative 8, and actually,

just for convenience, let's actually draw that
just for convenience, let's actually draw that

on the number line.
on the number line.

That's the number line.
That's the number line.

And if this is 5, x can't be equal to 5, so we draw a big
And if this is 5, x can't be equal to 5, so we draw a big

circle here, and then we would color in all the values
circle here, and then we would color in all the values

that x could be.
that x could be.

So x could be just the small-- it could be 5.0000001.
So x could be just the small-- it could be 5.0000001.

It just has to be a little bit bigger than 5, and any of those
It just has to be a little bit bigger than 5, and any of those

would satisfy it, right?
would satisfy it, right?

So let's just write some numbers that satisfy.
So let's just write some numbers that satisfy.

6 would satisfy it, 10 would satisfy it, 100
6 would satisfy it, 10 would satisfy it, 100

would satisfy it.
would satisfy it.

Now, if I were to multiply, or I guess divide, both sides of
Now, if I were to multiply, or I guess divide, both sides of

this, I guess we could say, equation or this inequality
this, I guess we could say, equation or this inequality

by negative 1, I want to understand what happens.
by negative 1, I want to understand what happens.

So what's the relation between negative x and negative 5?
So what's the relation between negative x and negative 5?

When I say what's the relation, is it greater than or is
When I say what's the relation, is it greater than or is

it less than negative 5?
it less than negative 5?

Well, 6 is a value that works for x.
Well, 6 is a value that works for x.

So negative 6, is that greater than or less than negative 5?
So negative 6, is that greater than or less than negative 5?

Well, negative 6 is less than negative 5, right?
Well, negative 6 is less than negative 5, right?

So let me draw the number line here.
So let me draw the number line here.

If we have negative 5 here, and let's just draw a circle around
If we have negative 5 here, and let's just draw a circle around

it because we know it's not going to be equal to negative 5
it because we know it's not going to be equal to negative 5

because we're deciding between greater than or less than.
because we're deciding between greater than or less than.

So we're saying 6 works for x.
So we're saying 6 works for x.

So negative 6 is here, right?
So negative 6 is here, right?

So negative 6 is less than negative 5, so is negative 10,
So negative 6 is less than negative 5, so is negative 10,

so is negative 100, so is negative a million, right?
so is negative 100, so is negative a million, right?

So it turns out that negative x is less than negative 5.
So it turns out that negative x is less than negative 5.

And this is really all you have to remember when you are
And this is really all you have to remember when you are

working with inequalities in algebra.
working with inequalities in algebra.

Inequalities, you can treat them just the way-- a greater
Inequalities, you can treat them just the way-- a greater

than or less than sign, you could treat them exactly the
than or less than sign, you could treat them exactly the

way you would treat an equal sign.
way you would treat an equal sign.

The only difference is, if you multiply or divide both sides
The only difference is, if you multiply or divide both sides

of the equation by a negative number, you swap it.
of the equation by a negative number, you swap it.

That's all you have to remember.
That's all you have to remember.

Let's do some problems, and hopefully, that'll
Let's do some problems, and hopefully, that'll

hit the point home.
hit the point home.

And if you ever forget, you just have to try-- you just
And if you ever forget, you just have to try-- you just

remember this: if x is greater than 5, well, then negative
remember this: if x is greater than 5, well, then negative

x is less than negative 5.
x is less than negative 5.

And keep trying out numbers.
And keep trying out numbers.

That's what's going to give you the best intuition.
That's what's going to give you the best intuition.

Let's do some problems.
Let's do some problems.

So if I said that 3x plus 2 is, let's say, less than or equal
So if I said that 3x plus 2 is, let's say, less than or equal

to 1-- well, this is a pretty easy equation to solve.
to 1-- well, this is a pretty easy equation to solve.

We just say 3x-- let's subtract 2 from both sides, and when you
We just say 3x-- let's subtract 2 from both sides, and when you

add or subtract, you don't do anything to the inequality.
add or subtract, you don't do anything to the inequality.

So if you subtract 2 from both sides, you get 3x is less than
So if you subtract 2 from both sides, you get 3x is less than

or equal to negative 1, right?
or equal to negative 1, right?

And then, now we're going to divide both sides by 3.
And then, now we're going to divide both sides by 3.

We get x is less than or equal to negative 1/3, right?
We get x is less than or equal to negative 1/3, right?

And notice, we didn't change anything because
And notice, we didn't change anything because

we divided both sides by a positive 3, right?
we divided both sides by a positive 3, right?

We could have actually done this equation in a
We could have actually done this equation in a

slightly different way.
slightly different way.

What if we subtracted 1 from both sides?
What if we subtracted 1 from both sides?

So this is another way of solving it.
So this is another way of solving it.

What if we said 3x plus 1 is less than or equal to 0, right?
What if we said 3x plus 1 is less than or equal to 0, right?

I just subtracted 1 from both sides, and now let's subtract
I just subtracted 1 from both sides, and now let's subtract

3x from both sides.
3x from both sides.

And we get 1 is less than or equal to minus 3x, right?
And we get 1 is less than or equal to minus 3x, right?

I subtracted 3x from here, so I have to subtract 3x from here.
I subtracted 3x from here, so I have to subtract 3x from here.

Now, I would have to divide both sides by a
Now, I would have to divide both sides by a

negative number, right?
negative number, right?

Because I'm going to divide both sides by negative 3.
Because I'm going to divide both sides by negative 3.

So I get negative 1/3 on this side, and based on what we had
So I get negative 1/3 on this side, and based on what we had

just learned, since we're dividing by a negative number,
just learned, since we're dividing by a negative number,

we want to swap the inequality, right?
we want to swap the inequality, right?

It was less than or equal, now it's going to be
It was less than or equal, now it's going to be

greater than or equal to x.
greater than or equal to x.

Now, did we get the same answer when we did it
Now, did we get the same answer when we did it

both-- two different ways?
both-- two different ways?

Here, we got x is less than or equal to negative 1/3, and
Here, we got x is less than or equal to negative 1/3, and

here we got negative 1/3 is greater than or equal to x.
here we got negative 1/3 is greater than or equal to x.

Well, that's the same answer, right? x is less than or
Well, that's the same answer, right? x is less than or

equal to negative 1/3.
equal to negative 1/3.

And that's-- I always find that to be the cool
And that's-- I always find that to be the cool

thing about algebra.
thing about algebra.

You can tackle the problem in a bunch of different ways, and
You can tackle the problem in a bunch of different ways, and

you should always get to the right answer as long as
you should always get to the right answer as long as

you, I guess, do it right.
you, I guess, do it right.

Let's do a couple more problems.
Let's do a couple more problems.

Oh, let's erase this thing.
Oh, let's erase this thing.

There you go.
There you go.

I'll do a slightly harder one.
I'll do a slightly harder one.

Let's say negative 8x plus 7 is greater than 5x plus 2.
Let's say negative 8x plus 7 is greater than 5x plus 2.

Let's subtract 5x from both sides.
Let's subtract 5x from both sides.

Negative 13x plus 7 is greater than 2.
Negative 13x plus 7 is greater than 2.

Now, we could subtract 7 from both sides.
Now, we could subtract 7 from both sides.

Negative 13x is greater than minus 5.
Negative 13x is greater than minus 5.

Now, we're going to divide both sides of this
Now, we're going to divide both sides of this

equation by negative 13.
equation by negative 13.

Well, very easy.
Well, very easy.

It's just x, and on this side, negative 5 divided by
It's just x, and on this side, negative 5 divided by

negative 13 is 5/13, right?
negative 13 is 5/13, right?

The negatives cancel out.
The negatives cancel out.

And since we divided by a negative, we switch the sign.
And since we divided by a negative, we switch the sign.

x is less than 5/13.
x is less than 5/13.

And once again, just like the beginning, if you don't believe
And once again, just like the beginning, if you don't believe

me, try out some numbers.
me, try out some numbers.

And I remember when I first learned this, I didn't believe
And I remember when I first learned this, I didn't believe

the teacher, so I did try out numbers, and that's how I got
the teacher, so I did try out numbers, and that's how I got

convinced that it actually works.
convinced that it actually works.

When you multiply or divide both sides of this equation
When you multiply or divide both sides of this equation

by a negative sign, you swap the inequality.
by a negative sign, you swap the inequality.

And remember, that's only when you multiply or divide, not
And remember, that's only when you multiply or divide, not

when you add or subtract.
when you add or subtract.

I think that should give you a good idea of how
I think that should give you a good idea of how

to do these problems.
to do these problems.

There's really not much new here.
There's really not much new here.

You do an inequality-- or I guess you could call this an
You do an inequality-- or I guess you could call this an

inequality equation-- you do it exactly the same way you do a
inequality equation-- you do it exactly the same way you do a

normal linear equation, the only difference being is if you
normal linear equation, the only difference being is if you

multiply or you divide both sides of the equation by a
multiply or you divide both sides of the equation by a

negative number, then you swap the inequality.
negative number, then you swap the inequality.

I think you're ready now to try some practice problems.
I think you're ready now to try some practice problems.

Have fun.
Have fun.