[MUSIC] Let's learn how to solve Absolute Value Inequalities.
Let's solve the following inequality for X and put our answer in interval
notation. If the absolute value is < or = 9, that
means what's inside the absolute value is between -9 and 9.
In other words, we are going to use the following factor to help us.
If the absolute value of Y is < or = to a Means that -a <= y <= a.
Lets apply that here. Namely, -9 <= 5 minus 2 X is less than or
equal to 9. Now remember, when solving this type of
compound inequality, we can work with all the parts at the same time.
So let's start by subtracting 5 everywhere.
Which gives us negative 9 minus 5. <= 5 - 5 - 2x <= 9 - 5,
or -14 <= -2x <= 4. And now let's divide everything by
negative two, however remember when we divide by a negative, we better flip or
reverse the direction of both of these inequalities.
Which give us 7 >= x >= -2. Which we usually write in the other
direction, -2 <= x <= 7. Let's look at this on the number line.
Let's say this is -2 and this is 7. x is greater than or equal To negative 2,
so x can equal negative 2. And then we go to the right, however x
has to be less than or equal to 7, so we stop here.
And we're asked to put our answer in interval notation.
Therefore our answer would be closed bracket.
So we want to include -2, up to 7. Again, closed bracket, because we want to
include 7. Alright lets look at another example.
[SOUND] Lets solve this inequality, and again put our answer in interval
notation. Done.
Now the difference here is that we have greater than rather than less than like
the last example. Now if the absolute value is greater than
1, then what's inside the absolute value is either larger than 1 or smaller than
-1. In other words, we're going to use the
following fact to help us solve this problem.
That the absolute value of y greater than a means that y is greater than a or y is
smaller than negative a. And this is what we're going to apply
here. And this gives us that 3 + 2x is either >
1 or 3 + 2x < -1. And now, when solving this type of
compound inequality, we work with each side separately.
So let's start with this 1st inequality here.
We'll bring the 3 to the right or subtract 3 on both sides, which gives us
2x is greater than 1 minus 3 or 2x is greater than negative 2.
And now dividing both sides by 2. We get x is greater than -1.
And what about this 2nd inequality? Again we'll bring the 3 to the right-hand side,
which means that 2x is less than -1 minus 3.
Or 2x is less than -4. Again, dividing by 2, gives us x is less
than -2. And since these inequalities are joined
by the word or, then the solution is all x values that make at least one of these
True. Let's look at this on the number line.
Let's say that this is -2 and this is -1. Greater than -1, we do not want x to
equal -1, so we put an open circle. Greater than we go to the right.
X less than -2. Again, we put an open circle because X
can not equal -2. Less than, we go to the left.
But again we're asked to put our answer in interval notation.
And doing this gives us our answer of negative infinity up to -2 open
parentheses because we do not want x to equal -2.
union, again open parentheses -1 up to infinity.
Dr. Sarah EichhornAssociate Dean, Distance Learning & Tenured Lecturer
Dr. Rachel Cohen Lehman
