Absolute Value Equations
Discussion: Absolute value refers to the measure of distance from zero for any value on the number line.
For example, the absolute value of 3 is 3 (written as
) because there are three units between the
number 3 and zero on the number line. Negative 3 (-3) is also three units from 0, so its absolute value is
also 3 (written as
.)
-3
0
3
If I said to you, the absolute value of some unknown (call it x) is 3, stated mathematically as
would you conclude? In other words, what does x equal?
, what
Let’s work on a problem to see these ideas at work. If we know that 24 is the absolute value of the
number that is 4 less than x, what are the possible values for x? First step is to restate the English
sentence into a mathematical one.
It’s pretty easy to see that in the equation above x might equal 28. But isn’t there a second answer here?
The absolute value bars are “neutralizing” a possible negative sign on the side containing the absolute
value bars so this problem actually must be solved twice.
Solve:
One way:
Second way:
Graphically we would represent this as shown below:
20
0
28
General Rule: If the value of the absolute is greater than 0, there must be two solutions.
Do we have the same dilemma with this problem?
NO! Zero is never negative.
Solve:
=-4
General Rule: If the value of the absolute is equal to 0, there is only one solution.
What about this one?
EEk! If the value of the absolute is the measure of the distance to
zero, it can never be a negative number (we don’t count with negative numbers do we?) so there are no
real solutions to this problem.
General Rule: If the value of the absolute is less than 0, there are no real number solutions.
To Summarize these ideas:
Practice: Solve the following absolute value problems for all possible solutions.
1)
2)
3)
4)
5)
6)
Absolute Value Inequalities:
In equations, we find exact values for the unknown;
means x is exactly 2 units away from zero. On
the other hand, inequalities express a range of values. An example from yesterday,
has as its
solutions all number greater than 2. We, also, found that we can represent this graphically. On the line
below, graph
.
If the inequality involves an absolute value, the sentence changes dramatically. The expression
states, “The distance from x to 0 is greater than 2 units on both sides of zero.” Stated another way, if the
absolute value of x is greater than two then x is more than 2 or less than -2 because both numbers have
an absolute value of 2. See this represented graphically below.
-2
0
2
Now, let’s work this out mathematically to convince ourselves this is a true statement.
Solve:
or
or
(divided both sides by -1)
Remember, the absolute value bars “hide” the fact that the value of the absolute might actually be
negative. We have to take this into account by solving twice, once for a possible positive value on the left
and once for the possible negative value.
Here’s a more complex example to reinforce what we’ve talked about up to now.
Solve:
or
or
or
or
Graph the solution we got form the above problem
But just when you think you’ve got this thing figured out, “they” throw another twist at you. Consider this
mathematical statement. The expression
states, “The distance from x to 0 is less than or equal to
2 on both sides of zero.” Stated another way, if the absolute value of x is less than or equal to 2 then x is
the region between -2 and 2. (Note: since this is a “less than or equal to” symbol both end points are
included.)
Again, let’s work this out mathematically to convince ourselves this is a true statement.
Solve:
or
or
(dividing both sides by negative 1)
However, we can state this more efficiently using a compound statement that looks just like the number
line. In fact, you should always restate this type of inequality as a compound statement.
Reading this mathematical expression from left to right we see that negative 2 is less than or equal to the
middle relation, x.
Actually there is a better way to do this. You can always rewrite the original absolute value problem of
this type as a compound inequality and solve it that way. Your answer will then automatically be in its
correct form. Let’s do one like this:
Solve:
(Rewrite as a compound inequality)
(Add three to all parts to isolate x)
To review these ideas consider the following table:
Sentence
Meaning
The distance from x to 0 is exactly
“a” units
The distance from x to 0 is greater
than “a” units.
Solution
or
or
The distance from x to 0 is less
than “a” units
Practice: Solve the following absolute value problems for all possible solutions and draw the related
graphs.
7)
8)
9)
10)
11)
12)
For #13, #14, and #15 choose a variable and write as an absolute value inequality that represents the set
of numbers on a number line.
13) All numbers no more than six units from zero.
14) All numbers at least 515 units from zero.
15) x is within 5 units of 2
Using Absolute Value to Solve Radical Inequalities
In R-2 we found that if x is positive or zero,
Let’s look at an example:
From this we see that for negative x,
. But what if x is negative?
.
. So for any real number,
. But this is exactly how we defined
, so for any real number x,
or
.
Solve:
This is the same as solving:
So,
is our solution.
Practice: Solve the inequality. Write your answers in both inequality and interval notation.
16)