Values to make absolute value inequality true

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- [Voiceover] We have three
inequalities here that involve

absolute values and then
below them we have potential

values for x,

and what I want to do in this
video is see which of these

potential values for x actually
make the inequality true.

So let's start with
this first one in green,

here on the left.

It says the absolute
value of x is less than

the absolute value of negative seven.

So let's think about
which of these x values

would make this true.

And before I even try out the x values,

let's see if we can
simplify this inequality.

Now the thing that might jump
out at you and encourage you

to try these on your own
before I work through them,

what might jump out at you is that we know

what the absolute value
of negative seven is.

The absolute value of
negative seven is how far

from zero is negative seven?

Well, it's seven to the left.

The distance from negative
seven to zero is seven,

or another way to think about it is

the absolute value of any
number is always going to be

the non-negative version of it.

So this right over here,
absolute value of negative seven

is just going to be seven.

So an equivalent inequality
would be that the

absolute value of x needs
to be less than seven.

So let's see if its true
for x equals negative eight.

So if x equals negative eight,

then wherever we see the x we
put a negative eight there.

So let's see, is the absolute
value of negative eight

less than seven?

Is that true?

Well, the absolute value of negative eight

is just going to be eight,

so is eight less than seven?

No, it's greater than seven.

So that, x equals negative eight does not

satisfy the inequality.

Now x equals negative two?

Well, wherever we see an x,
let's put a negative two.

So the absolute value
of negative two needs

to be less than seven.

What's the absolute value of negative two?

Well it's going to be positive two.

Is positive two less than seven?

Sure, two is less than seven.

So this, this is x equals negative two,

satisfies our inequality.

The absolute value of negative
two is going to be less than

the absolute value of negative seven.

Then, finally, x equals six.

So the absolute value is
the absolute value of six.

Once again, everywhere I see
the x I just put the six there.

X equals six.

We're going to say the
absolute value of six,

is that less than seven?

Well the absolute value of
six is once again, just six,

Six is six to the right of zero.

Is six less than seven?

Yeah, sure, six is less than seven.

So x equals six and x equals negative two

both satisfy the inequality.

Now let's do this one here
in this magenta color.

And once again, encourage
you to pause the video

and try to work through
it out on your own.

Let's try x equals negative four.

So if x equals negative four,

we're going to say the
absolute value of negative four

is greater than five.

Absolute value of negative four,

well that's just going to be four.

Is four greater than five?

No, four is less than five.

So that doesn't work.

Now, x equals three.

Everywhere we see the x,
replace it with a three.

The absolute value of three,

is that greater than five?

Absolute value of three is just three,

so is three greater than five?

No, three is less than five.

I think you're seeing, hopefully you're

getting the hang of it.

So finally, if x equals negative nine.

The absolute value of
negative nine, that would

need to be greater than five.

Absolute value of negative nine,

well that's just going
to be positive nine.

So it's just going to be nine,

and is that greater than five?

Well, sure, nine is greater than five.

So x equals negative nine
satisfies the inequality.

All right, now let's do these ones

in this light purple color.

The absolute value of x
needs to be greater than

negative 16.

So there's something very
interesting about this one.

We don't even have to
look at the choices here.

Can you think of any x for
which this would not be true?

Well just think about it a little bit.

The absolute value of a number,

is it ever going to be negative?

No, the absolute value
of a number is going

to be zero or positive.

It's going to be non-negative,

so this right over here
is zero or positive,

or we could call that non-negative.

Zero or positive.

So if this thing over
here is zero or positive,

something that is zero or greater,

something that is zero
or positive is always

going to be greater
than a negative number.

So this is actually true for all x's.

We don't even have to try them out.

We could try them out,
actually we will try them out,

just so you see that,

but it's going to be true for all x's,

because when you take the absolute value,

if you take, if x is zero
it's going to be zero,

but for any other value,
any non-zero value of x,

the absolute value of it
is going to be positive.

And let's just see that,
we could put any x there,

and this statement's going to be true.

Absolute value of x equals negative 15.

Well, the absolute value of negative 15,

is that greater than negative 16?

Well, the absolute value of negative 15

is positive 15, and of course positive 15

is going to be greater than negative 16.

A positive is always going to
be greater than a negative.

So this is true.

If x equals three,
absolute value of three,

is that greater than negative 16?

Oh, whoops let me write it that way.

Absolute value of three, is
that greater than negative 16?

Sure, absolute value of three is three.

And three is positive, so it's going to be

greater than a negative number.

So that works.

And as I said, any x would work there.

And finally if x equals nine.

Well, absolute value of
nine, if x equals nine,

is that greater than negative 16?

Well sure, that's just nine,

and that is greater than negative 16.

Even if x was zero, then
you would have zero is

greater than negative 16,
which is absolutely true.

So any x here works.