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(male narrator) In this video,
we will look at absolute value inequalities,
where we must first isolate the absolute value.
That is to say, before we can set up our compound inequality,
we must first isolate...
the absolute value.
Beware as we do this, with absolute value,
we cannot distribute...
or combine...
unlike terms.
We cannot...and by unlike terms, I mean,
if one term has an absolute value and one term does not,
they cannot be combined.
For example, in this problem,
before we can set up our tripartite inequality,
or before we can set up our "OR" inequality,
we must first isolate the absolute value.
What we cannot do
is distribute the 7 through the absolute value.
We also cannot combine the 2 and -7,
because the 7 is attached to an absolute value--
the 2 is not.
So we begin isolating just like solving any equation
by subtracting the 2 from both sides.
This gives us -7 times the absolute value of 3x plus 4
is less than -21.
Now, by dividing both sides by -7,
because we have -7 times the absolute value,
we can get rid of the -7--
leaving the absolute value of 3x plus 4.
Notice we've divided by a negative,
so we flip the symbol "is greater than" 3.
Greater tells us we must set up an OR inequality.
We take the inside-- 3x plus 4--
and make it less than the -3.
OR...the 3x plus 4 must be greater than the +3.
And now, we can solve these compound inequalities.
First, we subtract 4 from both sides,
and 3x is less than -7.
Divide by 3, and x is less than -7/3.
OR solving the other equation, we subtract 4,
and we get 3x is greater than -1.
And then, we divide both sides by 3
to get x is greater than -1/3.
Again, we can put this
on our number line...
knowing that the OR inequality
that comes from absolute value
is going to go outside
of our two points.
From -7/3, we need to go down.
From -1/3, we need to go up.
And this becomes our graph.
We can represent this
in interval notation--
going down to negative infinity;
up to -7/3; curve bracket,
because it's just greater than
and less than;
union, to show the second part;
picking up again at -1/3;
up to infinity.
Let's try one more example
where we must first isolate the absolute value
before we set up the compound inequality.
Here, we subtract 5 from both sides
to get 2 times the absolute value
of 4x minus 1 is less than or equal to 12.
Again, to clear the 2, we must divide both sides by 2
to get the absolute value of 4x minus 1
is less than or equal to 6.
Because the absolute value is less than or equal to,
we will set up a tripartite inequality--
putting the 4x minus 1 between -6 and +6.
Adding 1 to all three parts
gives us -5 is less than or equal to 4x,
which is less than or equal to 7.
Dividing all three parts by 4
gives us -5/4 is less than or equal to x,
which is less than or equal to 7/4.
The tripartite inequality
puts the values
between the -5/4 and the +7/4.
Because this inequality
is "or equal to,"
we will use closed dots
at each point
and then connect the dots.
In interval notation,
with square brackets
to show the "or equal to;"
-5/4, 7/4;
again with a square bracket.
Isolate the absolute, then solve.