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(male narrator) In this video,
we will begin looking
at tripartite compound inequalities.
A tripartite inequality is an inequality with three parts.
As we balance these three-part inequalities,
it will be important to balance on all three parts,
leaving no part left off.
If something happens in the center,
we also do it to the left and the right,
as we work to isolate our variable.
For example, in this problem,
the variable's in the middle
with other stuff between the -5 and the 9.
We will begin balancing, like always, by adding 17,
trying to get the x alone in the middle.
As we do so,
we must balance on all three parts of this inequality--
meaning we add 17 on the right,
and we'll add the 17 on the left.
We now have 12 is less than or equal to 2x,
which is less than 26.
Finally, to get the variable alone in the middle,
we can divide by 2,
and we must divide by 2 on all three parts.
As we do, we get 6 is less than or equal to x,
which is less than 13.
With these tripartite inequalities,
once we've isolated the variable,
it should be quite simple to graph the result.
Because what we're saying
is the x is between the number 6 and 13.
Because it's "or equal to" 6,
we will have a closed dot at 6...
and an open dot at 13...
and we want to be between those two values.
In interval notation,
a square bracket at the low point--6--up to 13,
which is a curved bracket, because it's strictly less than.
Let's take a look at another example
where we solve a tripartite inequality.
In this problem, again,
we will balance all three parts by first subtracting 4,
working to get the x alone in the middle.
As we do, notice the left side does not disappear:
4 minus 4 is 0.
We still need a number on the left side.
With a three-part inequality,
you should always have three parts at every step:
7 minus 4 is 3.
And now, we can isolate the x
by dividing all three parts, even the 0, by -3.
Notice, we've divided by a negative,
which will require
our inequality symbols to flip directions.
We now have 0 is greater than x,
which is greater than or equal to -1.
Generally, with tripartite inequalities,
we like the inequality symbols to open the other direction.
So let's flip this completely over--
as if holding up a mirror--
putting the -1 on the left,
is less than or equal to x, which is less than 0.
And that way, we can see the values going
from smallest to greatest.
Let's put this on our number line now.
Again, with the tripartite inequality...
we know we're gonna be between the two numbers.
At -1, we need a closed dot, because it's "or equal to."
At 0, we need an open dot, because it's strictly less than.
And the x is in between them... in between those two values.
We can express this
in interval notation,
with a square bracket
showing or equal to -1;
and a curved bracket at 0,
the larger value.
And that completes our problem.
Tripartite inequalities simply have three parts,
and we must balance all three parts
as we isolate the variable in the middle.