## Inequalities

Inequalities simply represent the relationship between two quantities that are not equal to one another. Inequalities arise when one expression is greater than or less than another expression. ie. A $\gt$ B or A $\ge$ B and B $\lt$ A or B $\le$ A. In CAT and other entrance tests, about 2 to 3 questions in inequalities appear every year.

### 1. Symbols & Representations

The basic types of inequalities between two items are enumerated below.

Representation Meaning
$a \gt b$ $a$ is greater than $b$
$a \lt b$ $a$ is less than $b$
$a \ge b$ $a$ is greater than or equal to $b$
$a \le b$ $a$ is less than or equal to $b$
$a \ne b$ $a$ is not equal to $b$

Common ways of writing the range of values where the inequalities hold good are as follows.

Representation Meaning
(3, 4) All real numbers between 3 and 4, but not including 3 and 4.
[3, 4] All real numbers between 3 and 4 including 3 and 4.
[3, 4) All real numbers between 3 and 4, including 3 but not including 4.
(3, 4] All real numbers between 3 and 4, not including 3 but including 4.

### 2. Arithmetic Operations on Inequalities

Addition and Subtraction: Where $k$ is any real number, adding/subtracting $k$ on both sides of an inequation leaves it unchanged. We can move variables from one side of the inequation to the other using addition/subtraction.

Multiplication & Division: Where $k$ is a real number and $\bold{k \gt 0}$, then multiplying/dividing $k$ on both sides of an inequation leaves it unchanged. However, if $\bold{k \lt 0}$, then multiplying/dividing $k$ on both sides o an inequation, changes the sign of the inequality. We cannot move variables from one side of the inequation to the other using multiplication/division.

If then Example
$a \gt b$,
$k \in r$
$a + k \gt b + k$
$a - k \gt b - k$
$a - b \gt 0$ or $0 \gt b - a$
If $x \gt 5 + y \implies x - y \gt 5$
$\implies -5 \gt y - x \implies y - x \lt -5$
$a \gt b$
$k \gt 0, j \lt 0$
$ak \gt bk$
$aj \lt bj$
$2x - 10 \gt 2 \implies x - 5 \gt 1$
$a - b \gt -2 \implies \dfrac{a - b}{-2} \lt 1 \implies \dfrac{b - a}{2} \lt 1$
Note: $a \gt b$ cannot be written as $\dfrac{a}{b} \gt 1$ as we do not know if the variable $b$ is negative or positive.