For another tutorial on absolute value functions, go here. Definition.
What is  x  ?
 Click on the button above "click here to
start" and maximize the window obtained. The point corresponding to x
can be moved by dragging it.
 On the left panel of the applet select
"definition". Two points are on the number line: a point corresponding
to a real number a and a point corresponding to
a real number x which can be dragged.
 Use the slider on the left panel to set a
to 0 (if it is not already). On the main panel ( top left ) are displayed
x and  x .
 Now set x to 2, find the distance between the origin of
the number line (0) and x. Compare this distance to  x  shown
on the main panel.
 Now set x to 2, find the distance between the origin of
the number line (0) and x. Compare this distance to  x  shown
on the main panel.
 Repeat 3 and 4 above for x = 5 and x = 5 respectively.
 What do you conclude when you compare the absolute
values to the distances?
 Check that ( use other values for x if necessary)
 x  = x when x>= 0 and
 x  =  x when x < 0 What is  x  a  ?
 Use the slider on the left panel to set a
to 2 . On the main panel is displayed  x  a .

Set x to 4, find the distance between the point
corresponding to a and the point corresponding to x. Compare this distance
to  x  a  shown on the main panel.
 Set x to 0, find the distance between the point
corresponding to a and the point corresponding to x. Compare this distance
to  x  a  shown on the main panel.
 Repeat 3 and 4 above for x = 3 and x = 1 respectively.
 What do you conclude when you compare the absolute
values to the distances as defined above?
 Check that (use other values for x and a if necessary)
 x  a  = x  a when x >= a and
 x  a  = (x  a) when x < a Use the concept of distance to define  x  a . (Note
that  x  =  x  0 ). Use the Definition of the Absolute Value
Function to Explore Basic Equations and Inequalities of the form:
 x  a  =
b
 x  a  <
b
 x  a  > b On the left panel select "properties", set a
(the top slider) to 0 and set b to 1 (the lower slider). The main panel (middle
one) shows an equation
 x  = 1 (in dark green) and two inequalities
 x  < 1 (in
blue)
and
 x  > 1 (in
red)

The solution of  x  = 1 is shown (dark
green) as two points symmetric with respect to the origin: 1 and 1.
Explain the two solutions.
The solution set to  x  < 1 is given by
the blue interval (1,1). Use the fact that  x  is a distance to explain
the solution set.
The solution set to  x  > 1 is given by
the union of the intervals (infinity , 1) and (1,+infinity)
(in red). Use the fact that  x  is a distance to
explain the solution set.
 Set a to 1 and b to 3. You now have the equation
and inequalities
 x  1  =
3
 x  1  <
3
 x  1  > 3 Explain the two solutions of the equation and the solution
sets of the two inequalities. Use the results above and the definition to check that : A   x  a  = b
is equivalent to:
x  a
= b and x  a =  b B   x  a  <
b is equivalent
to:
b < x  a < b C   x  a  > b
is equivalent to:
x  a >
b or x  a < b Exercises: Answer the following questions
analytically and use the applet to check the answers. 1  Evaluate:
a)  2

b)  4  2  Solve the
equations: a)  x  =
0 b)  x  =
3 c)  x  2  = 1 3  Solve the inequalities:
a)  x  < 2 b)  x  1  >
3 c)  x + 2  < 2 Think about solving the following a)  x  = 2 b)  x  4 
<  4 