Section 43
Solving and Graphing Absolute Value Equations
The
absolute value
of a number is its distance from zero on a
number line
. For instance, 4 and –4 have the same absolute value (4):
So, the absolute value of a positive number is just the number itself, and the absolute value of a negative number is its opposite. The absolute value of 0 is 0. Easy!
The absolute value of
x
is written

x

. So,
4 = 4
–4 = 4
54221.997 = 54221.997
(–1/4) = 1/4
A FEW RULES TO REMEMBER:
The
absolute value of a product
is the same as the
product of the absolute values
. For instance:
(9)(–3) = 9–3 = (9)(3) = 27
(–11)(–10) = –11–10 = (11)(10) = 110

x
^{
3
}
y
 = 
x
^{
3
}

y
The same goes for
quotients
.
(10)/(–5) = 10/–5 = 10/5 = 2
However, the same thing doesn't always work for addition and subtraction!
–3 + 7 = 4 = 4
, but
–3 + 7 = 3 + 7 = 10
So be careful!
SOLVING ABSOLUTE VALUE EQUATIONS
To solve an absolute value equation, it is usually easiest to get the absolute value alone on one side. Then remove the absolute value signs and write two equations to be solved separately.
Example:
Solve for
x
.
3
x
– 2 = 5
Start by dividing both sides by 3.
Now, split the equation into parts, using the fact that if 
a
 =
b
, then either
a
=
b
or
a
=
–
b
.
Now solve both equations.
GRAPHING THE SOLUTIONS OF ABSOLUTE VALUE EQUATIONS AND INEQUALITIES
The graph of the solution of an absolute value equation will usually be two points on the number line. For example, the graph of the solution to the last example problem is:
The graph of the solution of an absolute value
inequality
will usually look like the graph of a
compound inequality
:
Example:
Solve for
x
.
3
x
– 2
5
This can be broken into a compound inequality (be careful about direction!):
The graph is shown below. Note that closed dots are used at the endpoints of the rays, since the inequalities are not "strict" ones.