Absolute value equation and Absolute value inequation
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 $x=2 \; or \; x =6$ Practice Questions
$\left|{x - 6}\right| = 7$
$\left|{x - 5}\right| = 9$
$\left|{x - 6}\right| = 8$
$\left|{x - 7}\right| = 2$
$\left|{x - 6}\right| = 7$
$\left|{x - 2}\right| = 1$
$\left|{x - 1}\right| = 9$
$\left|{x - 4}\right| = 1$
$\left|{x - 1}\right| = 10$
$\left|{x - 8}\right| = 2$
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)
Solved Examples
Question 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$ Question 2
$|x| < 2 $ Solution:
We know that Formula
|x| < a implies that -a< x< a
So -2 < x < 2 Question 3
$1< |x-2| < 4$ Solution:
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)
