In the final two sections of this chapter we want to
discuss solving equations and inequalities that contain absolute values. We will look at equations with absolute value in them
in this section and we’ll look at inequalities in the next section.
Before solving however we should first have a brief
discussion of just what absolute value is.
The notation for the absolute value of p is |p|.
Note as well that the absolute value bars are NOT parenthesis and in many cases don’t behave as parenthesis so be careful with
them.
There are two ways to define absolute value. There is a
geometric definition and a mathematical definition. We will look at both.
Geometric Definition
In this definition we are going to think of |p| as the distance of p
from the origin on a number line. Also we will always use a positive value for distance. Consider the
following number line.
From this we can get the following values of absolute
value.
All that we need to do is identify the point on the
number line and determine its distance from the origin. Note as well that we
also have |0|
= 0 .
Mathematical Definition
We can also give a strict mathematical/formula
definition for absolute value. It is,
This tell us to look at the sign of p and if
it’s positive we just drop the absolute value bar. If p is negative we drop the absolute value bars and then put in a negative in front of
it.
So, let’s see a couple of quick examples.
|4| = 4 because 4 ≥ 0
|– 8| = – ( – 8) 8 because – 8 < 0
|0| = 0 because 0 ≥ 0
|0| = 0 because 0 ≥ 0
Note
that these give exactly the same value as if we’d used the geometric
interpretation.
One
way to think of absolute value is that it takes a number and makes it positive.
In fact we can guarantee that,
| p | ≥ 0
regardless
of the value of p.
We
do need to be careful however to not misuse either of these definitions. For
example we can’t use the definition on,
| – x |
because
we don’t know the value of x.
Also,
don’t make the mistake of assuming that absolute value just makes all minus
signs into plus signs. In other words, don’t make the following mistake,
| 4x – 3 | ≠ 4x
+ 3
This
just isn’t true! If you aren’t sure that you believe that plug in a number for x.
For example if x = – 1 we would get,
7 =
| – 7| = |4( – 1) – 3| ≠ |4( –
1) + 3| = – 1|
There
are a couple of problems with this. First, the numbers are clearly not the same
and so that’s all we really need to prove that the two expressions aren’t the same. There is
also the fact however that the right number is negative and we will never get a negative
value out of an absolute value! That also will guarantee that these two expressions aren’t the
same.
The
definitions above are easy to apply if all we’ve got are numbers inside the
absolute value bars. However, once we put variables inside them we’ve got to start being very
careful.
It’s
now time to start thinking about how to solve equations that contain absolute
values. Let’s start off fairly simple and look at the following equation.
|p| = 4
Now,
if we think of this from a geometric point of view this means that whatever p
is it must have a distance of 4 from the origin. Well there are only two
numbers that have a distance of 4 from the origin, namely 4 and -4. So, there
are two solutions to this equation,
p =
– 4 or p
= 4
Now, if you think about it we can
do this for any positive number, not just 4. So, this leads to the following
general formula for equations involving absolute value.
If
|
|p| = b,
b > 0 then p = – b or p
= b
|
Notice
that this does require the b be a positive number. We will deal
with what happens if b is zero or negative in a bit.
So, summarizing we can see that if b is zero
then we can just drop the absolute value bars and solve the equation. Likewise, if b is negative then there will be no
solution to the equation.
To this point we’ve only looked at equations that
involve an absolute value being equal to a number, but there is no reason to think that there has to only be a number on
the other side of the equal sign.
Likewise, there is no reason to think that we can only have one
inequality in the problem. So, we need to take a look at a couple of these kinds of equations.
So, as we’ve seen in the previous set of examples we
need to be a little careful if there are variables on both sides of the equal sign. If one side does not contain an
absolute value then we need to look at the two potential answers and make sure that each is in fact a
solution.
Sumber
Sumber
Labels:
Mathematician
Thanks for reading Absolute Value Equations . Please share...!
