Absolute value inequalities

U05_L1_T2_we1 : Absolute value inequalities

We're told to graph all possible
values for h on the

number line.

And this is a especially
interesting inequality because

we also have an absolute
value here.

So the way we're going to do it,
we're going to solve this

inequality in terms of the
absolute value of h, and from

there we can solve it for h.

So let's just get the absolute
value of h on

one side of the equation.

So the easiest way to do this
is to add 19 and 1/2 to both

sides of this equation.

I often like putting that as an
improper fraction, but 1/2

is pretty easy to deal with,
so let's add 19 and 1/2 to

both sides of this inequality.

Did I just say equation?

It's an inequality, not an
equation, it's an inequality

sign, not an equal sign.

So plus 19 and 1/2.

On the left-hand side, these
guys obviously cancel out,

that was the whole point, and we
are left with the absolute

value of h on the left-hand
side is less than.

And then if we have 19 and 1/2,
essentially minus 12, 19

minus 12 is 7, so it's going
to be 7 and 1/2.

So now we have that the absolute
value of h is less

than 7 and 1/2.

So what does this tell us?

This means that the distance,
another way to interpret

this-- remember, absolute value
is the same thing as

distance from 0-- so another
way to interpret this

statement is that the distance
from h to 0 has to be less

than 7 and 1/2.

So what values of h are going
to have less than a distance

from 7 and 1/2?

Well, it could be less than 7
and 1/2 and greater than 0, or

equal to 0.

So let me put it this way.

So h could be less
than 7 and 1/2.

But if it gets too far negative,
if it goes to

negative 3, we're cool, negative
4, negative 5,

negative 6, negative 7, we're
still cool, but then at

negative 8, all of a sudden the
absolute value isn't going

to be less than this.

So it also has to be greater
than negative 7 and 1/2.

If you give me any number in
this interval, its absolute

value is going to be less than 7
and 1/2 because all of these

numbers are less than 7
and 1/2 away from 0.

Let me draw it on the number
line, which they

want us to do anyway.

So if this is the number line
right there, that is 0, and we

draw some points, let's say
that this is 7, that is 8,

that is negative 7, that
is negative 8.

What numbers are less than
7 and 1/2 away from 0?

Well, you have everything all
the way up to-- 7 and 1/2 is

exactly 7 and 1/2 away, so you
can't count that, so 7 and

1/2, you'll put a circle
around it.

Same thing true for negative 7
and 1/2, the absolute value,

it's exactly 7 and 1/2 away.

We have to be less than 7 and
1/2 away, so neither of those

points are going to be included,
positive 7 and 1/2

or negative 7 and 1/2.

Now, everything in between is
less than 7 and 1/2 away from

0, so everything else counts.

Everything outside of it is
clearly more than 7 and 1/2

away from 0.

And we're done.