Testing solutions to inequalities

- [Voiceover] We have two
inequalities here, the first one

says that x plus two is
less than or equal to two x.

This one over here in I guess
this light-purple-mauve color,

is three x plus four
is greater than five x.

Over here we have four numbers
and what I want to do in this

video is test whether
any of these four numbers

satisfy either of these inequalities.

I encourage you to pause this video and

try these numbers out, does
zero satisfy this inequality?

Does it satisfy this one?
Does one satisfy this one?

Does it satisfy that one?

I encourage you to try
these four numbers out

on these two inequalities.

Assuming you have tried that, let's work

through this together.

Let's say, if we try out
zero on this inequality

right over here, let's substitute

x with zero.

So, we'll have zero plus two needs to

be less than or equal to two times zero.

Is that true?

Well, on the left hand side,
this is two needs to be

less than or equal to zero.

Is that true, is two less
than or equal to zero?

No, two is larger than zero.

So this is not going to be
true, this does not satisfy

the left hand side
inequality, let's see if it

satisfies this inequality over here.

In order to satisfy it,
three times zero plus four

needs to be greater than five times zero.

Well three times zero is just
zero, five times zero is zero.

So four needs to be greater
than zero, which is true.

So it does satisfy

this inequality right over here so zero

does satisfy this inequality.

Let's try out one.

To satisfy this one, one plus
two needs to be less than

or equal to two.

One plus two is three, is three
less than or equal than two?

No, three is larger than two.

This does not satisfy
the left hand inequality.

What about the right hand
inequality right over here?

Three times one plus four
needs to be greater than

five times one.

So three times one is three, plus four.

So seven needs to be greater
than five, well that's true.

Both zero and one satisfy
three x plus four is greater

than five x, neither of them
satisfy x plus two is less than

or equal to two x.

Now let's go to the two.

I know it's getting a
little bit unaligned,

but I'll just do it all in the
same color so you can tell.

Let's try out two here, two
plus two needs to be less than

or equal to two times two.

Four needs to be less
than or equal to four.

Well four is equal to
four and it just has to be

less than or equal, so this satisfies.

This satisfies this inequality.

What about this purple inequality?

Let's see, three times
two plus four needs to be

greater than five times two.

Three times two is six plus four is ten,

needs to be greater than 10.

10 is equal to 10, it's
not greater than 10.

It does not satisfy this inequality.

If this was a greater than or equal to

it would have satisfied but it's not.

10 is not greater than 10.

It would satisfy greater than or equal to

because 10 is equal to 10.

So two satisfies the left hand one but not

the right hand one.

Let's try out five.

Five plus two needs to
be less than or equal

to two times five, once
again everywhere we see

an x, we replace it with a five.

Seven needs to be less
than or equal to 10.

Which is absolutely true,
seven is less than 10.

So it satisfies less than or equal to.

Five satisfies this
inequality and what you're

probably noticing now is
that an inequality can

have many numbers that satisfy.

In fact they sometimes will
have nothing that satisfies it

and sometimes they might
have an infinite number of

numbers that satisfy it and
you see that right over here.

We're just testing out a few numbers.

For this left one, zero
and one didn't work,

two and five did work.

This right one, zero and
one worked, two didn't work.

Let's see what five does.

In order for five to
satisfy it, three times x.

Now we're gonna try x being five.

Three times five plus four
needs to be greater than

five times five.

Three times five is fifteen,
fifteen plus four is nineteen.

Nineteen is to be greater
than 25, it is not.

So five does not satisfy

this inequality right over here.

Anyway, hopefully you found that fun.