## Solving Equations Involving Absolute Values

The absolute value of a number is its distance from the origin.
For every (non-zero) distance there are two numbers with that distance
from the origin, a positive and a negative one. ** That is the only
fact peculiar to absolute value equations that you need to understand
and appreciate.** All absolute value equations in this class can be approached
using this simple principle.

For example, if we know that

then either or
.
Let's consider more examples.
Suppose

There are two possibilities for the left hand side of this equation:

Similarly there are two possibilities for the right hand side of the
equation:
We can now combine the various cases and obtain four
equations:
Of these the fourth equation can be obtained from the first, and the
third from the second, by multiplying with on both sides of the
equation. Therefore we need to consider only the first two equations. The
first equation has the solution and the second
.
It's easy to check (and you should do so) that both solutions
satisfy the original equation.
The above example is typical: ** we look at all possible
sign combinations of the expressions of which we take absolute values,
and prune the set by looking for equivalent equations.**

Sometimes there are subtleties.
Here is another example:
Suppose

Considering different cases we again obtain two equations
and
The last equation has the solution
, which is also
a solution of the original equation. However, there is no number
such that . The original equation, therefore has only
one solution!
So we solve an absolute value equation by replacing it with
several equations that do not involve absolute values, and solving
each one of them. Sometime the replacement equations have solutions
that are not satisfied by the original equation. A simple example is
given by

which leads to the replacement equations and .
However, neither nor solve the original equations.
Of course in this example it is obvious that there is no solution
since the absolute value of anything is never negative. Another less
obvious example is given by

The following table
illustrates all possibilities
Two of the four equations have no solutions. Moreover, it's easy to
check (do it) that
and
do not satisfy the
original equation. Indeed, that equation has no solution.