Why are Inequalities Important?

Sometimes we do not know exactly the value of an unknown, but we know the boundaries of possible values. Long before having the knowledge of Irrational nos., people knew that the circumference to diameter ratio, \(\pi\) is greater than 3. And 3 is then the lower bound of \(\pi\). People then worked hard to narrow down the boundaries (both lower and upper bounds) of \(\pi\) and getting more and more accurate value of it though its exact value can neither be written as a fraction nor a decimal figure since it is an irrational no.

Certain inequalities are particularly important in algebra. Besides Linear Programming, concepts of inequalities are frequently used in other areas, like Graph of Functions, Geometric Sequences and Arithmetic Sequences, etc.

Elementary Properties

Given \( \color{Tomato}{\boxed{ \color{RoyalBlue}{a > b}}} \color{DimGray}{\, \text{, where } a,\, b \in \mathbb{R} } \\[0.5em]
\; \begin{aligned}  &a+x > b +x \quad \forall x \in \mathbb{R}  \quad &\color{RoyalBlue}{(1.1)} \\[0.5em]
\,  &ax > bx  \; \text{ when } x >0 \text{; but}  \\
&ax < bx  \;\text{ when } x <0 \quad  &\color{RoyalBlue}{(1.2)} \\[0.5em]
\, &\dfrac{1}{a}< \dfrac{1}{b} \; \text{when } ab>0; \text{but}\\[0.2em]
&\dfrac{1}{a} > \dfrac{1}{b} \; \text{when }ab<0 \quad &\color{RoyalBlue}{(1.3)} \end{aligned}\)

Please note that in (1.3), \(ab>0\) means they are of the same sign.

Example 1:
If \( 0<x<1,\\  0<y<1, \)
prove that
\(\;\; 0 < x+y-xy <1\)

Quadratic Inequalities

Considering a quadratic equation, if the equal sign is replaced by an inequality sign, then it becomes a Quadratic Inequality.

Steps of Solving Quadratic Inequalities:

Recall the Quadratic Formula:

\(\quad \color{Tomato}{ \boxed{ \tiny \phantom{xxx} \\ \; \color{RoyalBlue}{x = \dfrac{-b \pm \sqrt {b^2-4ac}}{2a} \; \\[.3em] }}}\)