# Definition of the Absolute Value

The definition and properties of the absolute value function are explored interactively using an applet. The properties of basic equations and inequalities with absolute value are included.

For another tutorial on absolute value functions, go here.

### Definition.

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What is | x | ?

1. 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.
2. 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.
3. 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 |.
4. 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.
5. 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.
6. Repeat 3 and 4 above for x = 5 and x = -5 respectively.
7. What do you conclude when you compare the absolute values to the distances?
8. Check that ( use other values for x if necessary)

| x | = x when x>= 0

and | x | = - x when x < 0

What is | x - a | ?

1. Use the slider on the left panel to set a to 2 . On the main panel is displayed | x  - a |.
2. 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.
3. 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.
4. Repeat 3 and 4 above for x = 3 and x = 1 respectively.
5. What do you conclude when you compare the absolute values to the distances as defined above?
6. 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)

1.  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.

2.   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