Solve absolute value inequalities

Introduction:

        Other than algebra, there is one more important inequality. It is called as Absolute value inequality. The absolute value for any number is numerical value regardless of its sign.  For example, the absolute value of | -5 | is 5 and │+5 │ is 5. Here the vertical lines denotes absolute value.

It is a little more difficult when dealing with equations. There are three possible outcomes for absolute value inequalities. 

1. If x is any expression and a any positive number, and │x │= a, then it has either x = a or x = - a values.
2. If x is any expression and a any positive number, and │x │<>
3. If x is any expression and a any positive number, and │x │> a then it looks   x < -a and x > a.

Example

Solve: │4x – 4 │ = 8

Solution:

Use the result, if |x| = a, then x = a or x = -a

Here │4x – 4 │ = 8

So, it has either 4x – 4 = 8 or  4x – 4 = -8 values.

Solve each equation using the addition and multiplication principles.

4x = 12,    4x = -4

after solving the equation, we get

x = 3,     x = -1. Answer

Hope you like the above explanation to solve the absolute value inequalities, Please leave your comments, if you have any doubts.