The Modulus function
This gives the absolute value, the value of a number without the sign. It is denoted with vertical bars - the modulus of x is |x|.
When solving equations involving the modulus function, it can be very useful to do a sketch. Pencil in the function which is the argument of the modulus function, then ink over all the places where this function is positive. For the places where the function is negative, draw the reflection in the x-axis.
To do the algebra, to solve your equation or inequality, divide the
x-axis into a number of ranges, and replace the modulus function by
either its argument or minus its argument, as follows:
|f(x)| = f(x) for all values of x where f(x)>0
|f(x)| = -f(x) for all values of x where f(x)<0
Then solve your equation or inequality in the normal way, but remember that you can only use solutions within the range of x for which this equation is valid.
A simple example: |x2-5| = 4
Since |f(x)| is either f(x) or -f(x), this is the same as solving
x2-5 = +/- 4
which gives x=-3, -1, 1 or 3.
See also: [modulusfnml]
Exercises: [modulusfnxl]
Specifications met: [modulusfnsl]