Math Absolute Value Inequalities

In math, the absolute value |x| of a real number x is x's arithmetical value lacking view to its symbol. So, for example, 86 is the absolute value of both 86 and −86. Generalization of the absolute value for real numbers occurs in a extensive diversity of math settings. Consider an absolute value is also definite for the complex numbers, the quaternions, prepared rings, fields and vector spaces. The absolute value is strictly associated to the ideas of magnitude, distance, and norm in different math and physical contexts.

Properties of the Math absolute value inequalities

The absolute value fundamental properties are:

 |x| = sqrt(x^2)                                      (1) Basic

 |x| \ge 0                                            (2)     Non-negativity

 |x| = 0 \iff x = 0                            (3)     Positive-definiteness

 |xy| = |x||y|\,                                  (4)     Multiplicativeness

 |x+y| \le |x| + |y|                            (5)     Subadditivity

Another important property of the absolute value includes these are the:

 |-x| = |x|\,                                      (6)     Symmetry

|x - y| = 0 \iff x = y                     (7)     Identity of indiscernible (equivalent to positive-definiteness)

|x - y| \le |x - z| +|z - y|                 (8)     Triangle inequality (equivalent to sub additivity)

|x/y| = |x| / |y| \mbox{ (if } y \ne 0) \,                   (9)      Preservation of division (equivalent to multiplicativeness)

|x-y| \ge ||x| - |y||                           (10)     (Equivalent to sub additivity)

Math absolute value inequalities – Examples:

Math absolute value inequalities – Example 1:

|x - 5| < 3

Set up the two times inequality -3 < x -5 < 3 and then solve.

-3 < x – 5 < 3

2 < x < 8

In interval notation, the answer is (2, 8).
Math absolute value inequalities – Example 2:

|2x + 3| <=-8

This solution will involve setting up two separate inequalities and solving each.

2x + 3 <= -8

2x <=-11

x<=-11/2

Else

2x+3 >=8

2x >= 5

x>=5/2

In interval notation, the answer is (-oo, -11/2) U (5/2, oo) .
Math absolute value inequalities – Example 3:

|x - 9| < 4

Set up the two times inequality -4 < x -9 < 4 and then solve.

-4 < x – 9 < 4

5 < x < 13

In interval notation, the answer is (5, 13).

Math absolute value inequalities – Example 4:

|3x + 3| <= - 8

This solution will involve setting up two separate inequalities and solving each.

3x + 3 <= -8

 3x <=-11

x<=-11/3

Else

3x+3 >=8

 3x >= 5

x>=5/3

In interval notation, the answer is (-oo, -11/3) U (5/3, oo).