Solving Inequalities

U05_L1_T1_we2 : Solving Inequalities

We have the inequality 2/3
is greater than negative

4y minus 8 and 1/3.

Now, the first thing I want to
do here, just because mixed

numbers bother me-- they're
actually hard to deal with

mathematically.

They're easy to think about--
oh, it's a little

bit more than 8.

Let's convert this to an
improper fraction.

So 8 and 1/3 is equal to-- the
denominator's going to be 3.

3 times 8 is 24, plus 1 is 25.

So this thing over here is the
same thing as 25 over 3.

Let me just rewrite
the whole thing.

So it's 2/3 is greater than
negative 4y minus 25 over 3.

Now, the next thing I want to
do, just because dealing with

fractions are a bit of a pain,
is multiply both sides of this

inequality by some
quantity that'll

eliminate the fractions.

And the easiest one I can think
of is multiply both

sides by 3.

That'll get rid of the 3's
in the denominator.

So let's multiply both sides
of this equation by 3.

That's the left-hand side.

And then I'm going to multiply
the right-hand side.

3, I'll put it in parentheses
like that.

Well, one point that I want to
point out is that I did not

have to swap the inequality
sign, because I multiplied

both sides by a positive
number.

If the 3 was a negative number,
if I multiplied both

sides by negative 3, or negative
1, or negative

whatever, I would have had to
swap the inequality sign.

Anyway, let's simplify this.

So the left-hand side, we have
3 times 2/3, which is just 2.

2 is greater than.

And then we can distribute
this 3.

3 times negative 4y
is negative 12y.

And then 3 times negative 25
over 3 is just negative 25.

Now, we want to get all of our
constant terms on one side of

the inequality and all of our
variable terms-- the only

variable here is y on the other
side-- the y is already

sitting here, so let's just get
this 25 on the other side

of the inequality.

And we can do that by
adding 25 to both

sides of this equation.

So let's add 25 to both sides
of this equation.

--Adding 25--

And with the left-hand side, 2
plus 25 is 27 and we're

going to get 27 is
greater than.

The right-hand side of the
inequality is negative 12y.

And then negative 25 plus 25,
those cancel out, that was the

whole point, so we're left
with 27 is greater than

negative 12y.

Now, to isolate the y, you can
either multiply both sides by

negative 1/12 or you could say
let's just divide both sides

by negative 12.

Now, because I'm multiplying
or dividing by a negative

number here, I'm going to need
to swap the inequality.

So let me write this.

If I divide both sides of this
equation by negative 12, then

it becomes 27 over negative 12
is less than-- I'm swapping

the inequality, let me do this
in a different color-- is less

than negative 12y over
negative 12.

Notice, when I divide both sides
of the inequality by a

negative number, I swap the
inequality, the greater than

becomes a less than.

When it was positive, I didn't
have to swap it.

So 27 divided by negative
12, well, they're both

divisible by 3.

So we're going to get, if we
divide the numerator and the

denominator by 3, we get
negative 9 over 4 is less

than-- these cancel out-- y.

So y is greater than negative
9/4, or negative 9/4

is less than y.

And if you wanted to write
that-- just let me write

this-- our answer is y is
greater than negative 9/4.

I just swapped the order, you
could say negative 9/4

is less than y.

Or if you want to visualize that
a little bit better, 9/4

is 2 and 1/4, so we could also
say y is greater than negative

2 and 1/4 if we want to put
it as a mixed number.

And if we wanted to graph it
on the number line-- let me

draw a number line right here,
a real simple one.

Maybe this is 0.

Negative 2 is right over, let's
say negative 1, negative

2, then say negative
3 is right there.

Negative 2 and 1/4 is going
to be right here, and it's

greater than, so we're not going
to include that in the

solution set.

So we're going to make an
open circle right there.

And everything larger than that
is a valid y, is a y that

will satisfy the inequality.