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Absolute value equation and Absolute value inequation for Class 11 ,CBSE Board, IITJEE maths and other exams





Absolute value equation:

Absolute value is denoted by |x|. And it is defined as

|x|  = x  if x $\geq$ 0
     =-x  if x < 0

So it is always positive.

Examples

|x-4|=2

or x-4=-2   or x-4=2
or x=2  or x =6


Absolute value inequation

|x-2| > 4
|x| < 2

This is a  form of Absolute value inequation

Important Formula's
for a and r being positive real number

|x| < a  implies that   -a< x< a
|x| > a  implies that   x< -a or  x> a

|x|   $\geq$  a implies that x $\geq$ a  or x $\leq$ a
|x-a|  < r implies that  a-r < x  < a+r

|x-a| > r   implies that  x < a-r  or x > a+r

a< |x| <  b  implies that x lies in (-b,-a)  or (a,b)
a< |x-c| < b implies that x lies in (-b+c,-a+c)  or (a+c,b+c)


Examples

1) |x-2| > 4
Solution
we know from the Formula
|x-a| > r   implies that  x < a-r  or x > a+r

So x < -2  or x > 6

2)  |x| < 2
Solution:
We know that Formula
|x| < a  implies that   -a< x< a
So  -2 < x < 2

 



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