## Solving Absolute Value Equations For Real Numbers

The safest way to solve absolute value equations for real numbers is to sketch the functions and label each section of each graph with a function. Any solutions can then be visualised and it is easy to see if a solution exists or does not. Example: Solve

$|x|=2|x-2|-1$
.

When we remove the modulus sign from an expression we can apply a

$+$
or a
$-$
option. Apply the
$+$
option to any section of graph going up and the
$-$
option to any section of graph going down/ For the graph
$2|x-2|-1$
we obtain
$2(x-2)-12x-5$
and
$2(-(x-2))-1=-2x+3$
.
We get

Now solve.

$x=2x-5 \rightarrow x=5$

$x=-2x+3 \rightarrow 3x=3 \rightarrow x=1$