Definition Properties Graphs of Functions Involving Modulus Inequalities Involving Modulus Links to Other Sites

efinition

The absolute value or modulus of a real number x, written |x| is defined as

The graph of y = |x| is shown below.

Geometrically, |x| is the distance of x from the origin.

Example 1:

|3 - x| =  ì í î 3 - x   -(3 - x)    if 3 - x ³ 0,   if 3 - x < 0;   =  ì í î 3 - x   x -3 if x £ 3,   if x > 3.

roperties

1. |-a| = |a|
2. Ö(a²) = |a|
3. |ab| = |a||b|
4. |a/b| = |a|/|b|  provided b ¹ 0
5. |a + b| £ |a| + |b|
6. |a - b| ³ |a| - |b|
7. For k ³ 0, |a| = k  Þ  a = ±k
8. For any k, |a| = |k|  Þ  a = ±k
9. For k > 0, |a| < k  Û  -k < a < k
10. For k > 0, |a| > k  Û  x < -k or x > k

raphs of Functions Involving Modulus

Since |f(x)| =

ì í î

f(x)   -f(x)

if f(x) ³ 0, if f(x) < 0;

the graph of y = |f(x)| is obtained from the graph of y = f(x) by reflecting the negative part (ie below the x-axis) about the x-axis.

Example 2:  Sketch y = |x² - 1|.

Example 3:  Sketch y = |x| + |2x - 3|.

x £ 0 0 £ x £ 3/2 x ³ 3/2 |x| -x x x |2x - 3| -2x + 3 -2x + 3 2x - 3 y = |x| + |2x - 3| -3x + 3 -x + 3 3x - 3 Example 6 will make use of this graph.

nequalities Involving Modulus

Example 4:  Solve |3x - 2| < 3 - 2x.

Solution:

Example 5:  Solve |3x - 2| > |3 - 2x|.

Solution:

Example 6:  Find the solution set of |x| + |2x - 3| ³ 4x.

Solution:

We have sketch the graph in Example 3. We will sketch an addition graph of y = 4x. At the point of intersection, -x + 3  = 4x x  = 3/5 The solution set is {x Î R : x £ 3/5}.

Example 7:  Solve

|x| + 1 ¾¾¾ |x| - 1

< 3.

Solution:

|x| + 1 - 3|x| + 3 ¾¾¾¾¾¾¾ |x| - 1  < 0   2|x| - 4 ¾¾¾ |x| - 1  > 0   Let y = |x|.  Since 2 is positive, we may ignored it. y - 2 ¾¾¾ y - 1  > 0

Thus  y < 1   or  y > 2 Þ    |x| < 1 or |x| > 2 -1 < x < 1 or x < -2 or x > 2 \  x < -2, -1 < x < 1 or x > 2

to Other Sites

• Absolute Value and Distance
• Absolute Value Inequalities