


Absolute Value
To solve an equation in which the variable is within absolute
value brackets, you must divide the equation into two equations.
The most basic example of this is an equation of the form
x = c. In this case, either x = c or x =
–c.
A slightly more complicated example is this:
In this problem, you must solve two equations: First,
solve for x in the equation x +
3 = 5. In this case, x = 2. Then, solve for x in
the equation x + 3 = –5. In this case, x =
–8. So the solutions to the equation x + 3 =
5 are x = {–8, 2}.
Generally speaking, to solve an equation in which the
variable is within absolute value brackets, first isolate the expression
within the absolute value brackets and then divide the equation
into two. Keep one of these two equations the same, while in the
other negate one side of the equation. In either case, the absolute
value of the expression within brackets will be the same. This is
why there are always two solutions to absolute value problems (unless the
variable is equal to 0).
Here is one more example:

First, isolate the expression within the absolute value
brackets:
Then solve for the variable as if the
expression within absolute value brackets were positive:
Next, solve for the variable as if the
expression within absolute value brackets were negative:
The solution set for x is {y^{2} –
3, –y^{2} –1}.
