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

$ =-x$ if $x < 0 $

So it is always positive.

$|x-4|=2$

or $x-4=-2 $ or $ x-4=2 $

or $x=2 \; or \; x =6$

- $\left|{x - 6}\right| = 7$
- $\left|{x - 5}\right| = 9$
- $\left|{x - 6}\right| = 8$
- $\left|{x - 7}\right| = 2$
- $\left|{x - 6}\right| = 7$
- $\left|{x - 2}\right| = 1$
- $\left|{x - 1}\right| = 9$
- $\left|{x - 4}\right| = 1$
- $\left|{x - 1}\right| = 10$
- $\left|{x - 8}\right| = 2$

$|x| < 2$

This is a form of Absolute value inequation

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)

$|x-2| > 4 $

we know from the Formula

$|x-a| > r$ implies that $x < a-r \; or \; x > a+r$

So $ x < -2 \; or \; x > 6$

$|x| < 2 $

We know that Formula

|x| < a implies that -a< x< a

So -2 < x < 2

$1< |x-2| < 4$

We know that Formula

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

So x lies (-4+2,-1+2) or (1+2,4+2)

or x lies (-2,1) or (3,6)

**Notes**- What are inequalities
- Things which changes the direction of the inequality
- Linear Inequation in One Variable
- Linear Inequation in Two Variable
- Steps to solve the inequalities in one variable
- Steps to solve the inequality of the another form
- Quadratic Inequation
- Steps to solve Quadratic or polynomial inequalities
- Cubic Inequation
- Steps to solve Cubic inequalities
- Absolute value equation
- Absolute value inequation
- Graphical Solution of Linear inequalities in Two Variable

**NCERT Solutions**