Absolute Value
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1. Evaluate
and
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To do these evaluations we need to
remember the definition of absolute value.
With this definition the evaluations are easy.
Remember that absolute value takes
any nonzero number and makes sure that it’s positive.
2. Eliminate
the absolute value bars from
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This one is a little different
from the first example. We first need to
address a very common mistake with these.
Absolute value doesn’t just change
all minus signs to plus signs. Remember
that absolute value takes a number and makes sure that it’s positive or zero. To convince yourself of this try plugging in
a number, say
There are two things wrong with
this. First, is the fact that the two
numbers aren’t even close to being the same so clearly it can’t be
correct. Also note that if absolute value
is supposed to make nonzero numbers positive how can it be that we got a -77 of out of
it? Either one of these should show you
that this isn’t correct, but together they show real problems with doing this,
so don’t do it!
That doesn’t mean that we can’t
eliminate the absolute value bars however.
We just need to figure out what values of x will give positive numbers and what values of x will give negative numbers. Once we know this we can eliminate the
absolute value bars. First notice the
following (you do remember how to Solve Inequalities
right?)
So, if then and if then . With this information we can now eliminate
the absolute value bars.
Or,
So, we can still eliminate the
absolute value bars but we end up with two different formulas and the formula
that we will use will depend upon what value of x that we’ve got.
On occasion you will be asked to
do this kind of thing in a calculus class so it’s important that you can do
this when the time comes around.
3. List
as many of the properties of absolute value as you can.
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Here are a couple of basic
properties of absolute value.
These should make some sense. The first is simply restating the results of
the definition of absolute value. In
other words, absolute value makes sure the result is positive or zero (if p =
0). The second is also a result of the
definition. Since taking absolute value
results in a positive quantity or zero it won’t matter if there is a minus sign in
there or not.
We can use absolute value with
products and quotients as follows
Notice that I didn’t include sums (or
differences) here. That is because in
general
To convince yourself of this consider
the following example
Clearly the two aren’t equal. This does lead to something that is often
called the triangle inequality. The
triangle inequality is
The triangle inequality isn’t used all that often in a Calculus course, but
it’s a nice property of absolute value so I thought I’d include it.