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 
. Note as well that the absolute value bars are
NOT parentheses and in many cases don’t behave as parentheses 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 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 .
Mathematical
Definition
We can also give a strict mathematical/formula definition
for absolute value. It is,
This tells 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.
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,
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
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,
This just isn’t true!
If you aren’t sure that you believe that plug in a number for x.
For example if we would get,
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.
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,
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.
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.
Let’s take a look at some examples.
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Example 1 Solve
each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
Now, remember that absolute value does not just make all
minus signs into plus signs! To solve
these we’ve got to use the formula above since in all cases the number on the
right side of the equal sign is positive.
(a)
There really isn’t much to do here other than using the
formula from above as noted above. All
we need to note is that in the formula above p represents whatever is on the inside
of the absolute value bars and so in this case we have,
At this point we’ve got two linear equations that are easy
to solve.
So, we’ve got two solutions to the equation and .
[Return to Problems]
(b)
This one is pretty much the same as the previous part so
we won’t put as much detail into this one.
The two solutions to this equation are and .
[Return to Problems]
(c)
Again, not much more to this one.
In this case the two solutions are and .
[Return to Problems]
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Now, let’s take a look at how to deal with equations for
which b is zero or negative. We’ll do this with an example.
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Example 2 Solve
each of the following.
(a)
(b)
Solution
(a) Let’s
approach this one from a geometric standpoint. This is saying that the quantity in the
absolute value bars has a distance of zero from the origin. There is only one number that has the
property and that is zero itself. So,
we must have,
In this case we get a single solution.
(b) Now, in
this case let’s recall that we noted at the start of this section that . In other words, we can’t get a negative
value out of the absolute value. That
is exactly what this equation is saying however. Since this isn’t possible that means there
is no solution to this equation.
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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 absolute value in the problem.
So, we need to take a look at a couple of these kinds of equations.
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Example 3 Solve
each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
At first glance the formula we used above
will do us no good here. It requires
the right side of the equation to be a positive number. It turns out that we can still use it here,
but we’re going to have to be careful with the answers as using this formula
will, on occasion introduce an incorrect answer. So, while we can use the formula we’ll need
to make sure we check our solutions to see if they really work.
(a)
So, we’ll start off using the formula above as we have in
the previous problems and solving the two linear equations.
Okay, we’ve got two potential answers here. There is a problem with the second one
however. If we plug this one into the
equation we get,
We get the same
number on each side but with opposite signs.
This will happen on occasion when we solve this kind of equation with
absolute values. Note that we really
didn’t need to plug the solution into the whole equation here. All we needed to do was check the portion
without the absolute value and if it was negative then the potential solution
will NOT in fact be a solution and if it’s positive or zero it will be solution.
We’ll leave it
to you to verify that the first potential solution does in fact work and so
there is a single solution to this equation : .
[Return to Problems]
(b)
This one will work in pretty much the same way so we won’t
put in quite as much explanation.
Now, before we
check each of these we should give a quick warning. Do not make the assumption that because the
first potential solution is negative it won’t be a solution. We only exclude a potential solution if it
makes the portion without absolute value bars negative. In this case both potential solutions will
make the portion without absolute value bars positive and so both are in fact
solutions.
So in this case, unlike the first example, we get two solutions
: and .
[Return to Problems]
(c)
This case looks
very different from any of the previous problems we’ve worked to this point
and in this case the formula we’ve been using doesn’t really work at
all. However, if we think about this a
little we can see that we’ll still do something similar here to get a
solution.
Both sides of the equation contain absolute values
and so the only way the two sides are equal will be if the two quantities
inside the absolute value bars are equal or equal but with opposite
signs. Or in other words, we must
have,
Now, we won’t need to verify our solutions here as we did
in the previous two parts of this problem.
Both with be solutions provided we solved the two equations
correctly. However, it will probably
be a good idea to verify them anyway just to show that the solution technique
we used here really did work properly.
Let’s first check .
In the case the
quantities inside the absolute value were the same number but opposite
signs. However, upon taking the
absolute value we got the same number and so is a solution. Now, let’s check .
In the case we
got the same value inside the absolute value bars.
So, as
suggested above both answers did in fact work and both are solutions to the
equation.
[Return to Problems]
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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.