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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
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