The Absolute Value


In my previous tutorial on MAT206[Intorduction to Real Analysis], i gave an introduction to the real number system with the algebraic structure in focus. Click here to redirect back to my last lecture on real numbers.
Studying this real number system, we will really find so many  interesting things expecially some of the properties this numbers tends to exhibit.
Another beautiful property the real number exhibit is its ability make a number or value "absolute". But the term absolute is a term that causes lots of confusion
as so many students tend to give their definitions on it base on what they all think.
I remember my days in high school when we define an absolute value as simply eliminating the negative sign of any given value.
But today i will clearing doubts on this problem. first i begin with basic definition of the absolute value and then give some examples.

An absolute value or otherwise known as the modulo $|x|$ of a real number $x$ is the non-negative value of $x$ without regard to the sign. for example we say $|x|=x$ for a positive value of $x$. and $|x|=-x$ for a negative value of $x$ in which case we say $-x$ is positive.
Now lets take more examples:
$|0|=0$
$|3|=3$
$-3=3$

Absolute value of a number is its distance from zero.
For any real number $x$ the absolute value or modulus of $x$ is given by:
\begin{align}
|x|=\begin{cases}
x, \text{if}\ x\geq{0}\\
-x, \text{if}\ x<{0}
\end{cases}
\end{align}

Now we see that the absolute value of $x$ is always either positive or zero, but never negative.

Now i will list some special properties of the absolute values.
 
(1) $|x|=\sqrt{a^2}$
(2) $|x|\geq{0}$ i.e $x$ is nonnegative.
(3) $|x|=0\Leftrightarrow{x=0}$
(4) $|xy|=|x||y|$
(5) $|x+y|\leq{|x|+|y|}$
(6) $||x||=|x|$ this is the norm property that gives rise to absolute value.
(7) $|-x|=|x|$
(8) $|x-y|=0\Leftrightarrow{x=y}$
(9) $|x-y|\leq{|x-z|+|z-y|}$ this is also called the triangular inequality.
(10) $\|frac{x}{y}|=|\frac{x}{y}|$ if and only if $y\neq{0}$
(11) $|x-y|\geq{||x|-|y||}$
(12) $|x|\leq{y}\Leftrightarrow{-y\leq{x}\leq{y}}$
(13) $|x|\geq{y}\Leftrightarrow{x\leq{-y}\ \text{or}\ y\leq{x}}$

Examples 1

(1) Evaluate $|4x-3|\leq{11}$

Solution:

To evaluate this all we need to do is, add a positive $3$ to both sides. Because we need to simplify the L.H.S and do away with the absolute value.
$|4x-3+3|\leq{11+3}$
$={|4x|\leq{14}}$
$={4x}\leq{14}$
$\Rightarrow{x}=\frac{7}{2}$

Examples 2

$|x^2-4|<5$

Solution

Add $(+4)$ to both sides
$|x^2-4+4|\leq{5+4}$
$|x^2|\leq{9}$
$x^2\leq{9}$
$x\leq{\pm{3}}$

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