In the previous section we solved equations that contained
absolute values. In this section we want
to look at inequalities that contain absolute values. We will need to examine two separate cases.
Inequalities
Involving < and 
As we did with equations let’s start off by looking at a
fairly simple case.
This says that no matter what p is it must have a distance of no more than 4 from the
origin. This means that p must be somewhere in the range,
We could have a similar inequality with the < and get a
similar result.
In general we have the following formulas to use here,
Notice that this does require
b to be positive just as we did with
equations.
Let’s take a look at a couple of examples.
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Example 1 Solve
each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
(a)
There really isn’t much to do other than plug into the
formula. As with equations p simply represents whatever is inside
the absolute value bars. So, with this
first one we have,
Now, this is nothing more than a fairly simple double
inequality to solve so let’s do that.
The interval notation for this solution is .
[Return to Problems]
(b)
Not much to do here.
The interval notation is .
[Return to Problems]
(c)
We’ll need to be a little careful with solving the double
inequality with this one, but other than that it is pretty much identical to
the previous two parts.
In the final step don’t forget to switch the direction of
the inequalities since we divided everything by a negative number. The interval notation for this solution is .
[Return to Problems]
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Inequalities
Involving > and
Once again let’s start off with a simple number example.
This says that whatever p
is it must be at least a distance of 4 from the origin and so p must be in one of the following two
ranges,
Before giving the general solution we need to address a
common mistake that students make with these types of problems. Many students try to combine these into a
single double inequality as follows,
While this may seem to make sense we can’t stress enough
that THIS IS NOT CORRECT!! Recall what a
double inequality says. In a double
inequality we require that both of the inequalities be satisfied
simultaneously. The double inequality
above would then mean that p is a
number that is simultaneously smaller than -4 and larger than 4. This just doesn’t make sense. There is no number that satisfies this.
These solutions must be written as two inequalities.
Here is the general formula for these.
Again, we will require
that b be a positive number
here. Let’s work a couple of examples.
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Example 2 Solve
each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
(a)
Again, p
represents the quantity inside the absolute value bars so all we need to do
here is plug into the formula and then solve the two linear inequalities.
The interval notation for these are or .
[Return to Problems]
(b)
Let’s just plug into the formulas and go here,
The interval notation for these are or .
[Return to Problems]
(c)
Again, not much to do here.
Notice that we had to switch the direction of the
inequalities when we divided by the negative number! The interval notation for these solutions
is or .
[Return to Problems]
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Okay, we next need to take a quick look at what happens if b is zero or negative. We’ll do these with a set of examples and
let’s start with zero.
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Example 3 Solve
each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
Solution
These four examples seem to cover all our bases.
(a) Now we know that and so can’t ever be less than zero. Therefore, in this case there is no
solution since it is impossible for an absolute value to be strictly less
than zero (i.e. negative).
(b) This is almost the same as the previous part. We still can’t have absolute value be less
than zero, however it can be equal to zero.
So, this will have a solution only if
and we know how to solve this from the previous section.
(c) In this case let’s again recall that no matter what p is we are guaranteed to have . This means that no matter what x is we can be assured that will be true since absolute values will
always be positive or zero.
The solution in this case is all real numbers, or all
possible values of x. In inequality notation this would be .
(d) This one is nearly identical to the previous part except this
time note that we don’t want the absolute value to ever be zero. So, we don’t care what value the absolute
value takes as long as it isn’t zero.
This means that we just need to avoid value(s) of x for which we get,
The solution in this case is all real numbers except .
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Now, let’s do a quick set of examples with negative numbers.
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Example 4 Solve
each of the following.
(a) and
(b) and
Solution
Notice that we’re working these in pairs, because this
time, unlike the previous set of examples the solutions will be the same for
each.
Both (all four?) of these will make use of the fact that
no matter what p is we are
guaranteed to have . In other words, absolute values are always
positive or zero.
(a) Okay, if
absolute values are always positive or zero there is no way they can be less
than or equal to a negative number.
Therefore, there is no solution for either of these.
(b) In this
case if the absolute value is positive or zero then it will always be greater
than or equal to a negative number.
The solution for each of these is then all real numbers.
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