Inequality is a relation created between two quantities. It is a method of comparing two values. The two values which are compared are not equal to each other, hence $a \neq b$. There are various symbols which are used to denote the relationship between the two quantities which are being compared. The symbols used are less than represented by ‘$<$’ symbol, greater than represented by ‘$>$’ symbol, less than or equal to denoted by ‘$\leq$’ and greater than or equal to denoted by ‘$\geq$’. When we solve an inequality equation we actually find out the solution of the variable. On plugin the solution of the variable in place of the variable it holds true for the inequality statement being considered.

## Properties of Inequalities

The properties of inequalities are as follows:

• Transitive Property - If inequalities are placed in order, then we can skip the inequality present in the middle. That is suppose $a < b$ and $b < c$, then $a < c$. Similarly, if $a > b$ and $b > c$, then $a > c$. For example if Ryan is older than John and John is older than Sam, then Ryan is older than Sam.

• Reversal Property - The inequality quantities being compared can be swapped or reversed provided the symbol still points at the smaller value. That is if $a > b$, then $b < a$. Similarly, if $a < b$, then $b > a$. For example if Ryan is older than John, it means John is younger than Ryan.

• Law of Trichotomy - It states that there are strictly three statements and one of them definitely holds true. The three statements are either $a$ is less than $b$ which is $a < b$ or a is equal to $b$ which is $a$ = $b$ or a is greater than $b$ which is $a > b$. For example, if Ryan has more marbles than John it means Ryan does not hold the same numbers of marbles like John or Ryan does not have less than marbles than John has.

• Addition and Subtraction - An inequality statement remains intact or unchanged if we either add or subtract the same value from both sides. That is if $a < b$ and we add $c$ on both sides, then $a + c < b + c$. Similarly if we subtract say $c$ from both sides, $a – c < b – c$ the inequality remains unaltered. Likewise, if $a > b$, then $a + c > b + c$ and $a – c > b – c$. For example Marks scored less marks than David. If three points are added to each of their scorecard still Mark would score less than David, the difference of score between them remains constant.

• Multiplication and Division - An inequality statement remains intact or unchanged if we multiply both sides of the inequality with the same positive number. But if we multiply with a negative number on both sides then the inequality gets swapped that is symbol reverses. That is if $a < b$ or $a > b$, and we multiply both sides with a positive value say $c$, then $ac < bc$ or $ac > bc$. But when we multiply both side of the inequality with a negative value say $d$, then $ad > bd$ and $ad < bd$. The symbol just reverses. Same holds true for division. For example score obtained by Mark is less than that of Joe. If their individual scores gets doubled then also the score obtained by Mark is less than that of Joe.

• Additive Inverse - If both sides of an inequality statement is prefixed with a minus symbol, the inequality symbol reverses. That is if $a < b$ then $–a > -b$. Similarly if $a > b$ then $–ac < -bc$

• Multiplicative Inverse - If the reciprocal of the two quantities of the inequality are taken then the symbol of inequality gets reversed. That is if $a < b$ or $a > b$, then $\frac{1}{a}$ > $\frac{1}{b}$ or $\frac{1}{a}$ < $\frac{1}{b}$

• Non Negative Property of Squares - Square of any real number is always greater than or equal to $0$. It is never a negative value. That is $a^{2} >= 0$

• Square Root Property - If square root is taken of the two quantities in the inequality statement then the relation remains unchanged. That is if $a < b$ or $a > b$ then $sqrt(a) < sqrt(b)$ or $sqrt(a) > sqrt(b)$. There is a condition to be maintained in it which is both $a$ and $b$ need to be positive values.

## How to Use Inequalities

Inequalities are used to compare two values. The symbols of inequality are placed in between the two quantities into consideration. Suppose if two values $a$ and $b$ are compared then either of the four inequality symbol ‘$<$’, ‘$>$’, ‘$<=$’, ‘$>=$’ are placed in between the two inequality quantities a and b being compared.

## Inequality Notation

Open interval and closed intervals are used to denote the inequality notation. That is $b > a > c$ which means if $a < b < c$, then the interval notation shall be $(a, c)$. If $a < b <=c$, then the interval notation shall be $(a, c]$. Open interval shows the less than greater than values but closed interval shows that value is inclusive. Similarly, if $a <= b <=c$, then the interval notation shall be $[a, c]$.

## Less than or Greater than

Symbols used to represent or compare the two quantities which are either less than, greater than, less than or equal to and greater than equal to are $‘<’,\ ‘>’.\ ‘<=’,\ ‘>=’$

## Examples

Example 1:

Solve: $12 < x + 5$

Solution:

We subtract $5$ from both sides, that is $12 - 5 < x + 5 - 5$.

Therefore, $7 < x$ which mean $x$ has values greater than $7$.

Interval notation ($7$, infinity)
Example 2:

Solve: $\frac{(x – 3)}{2}$ $< -5$

Solution:

Multiplying both sides with $2$,

$(x – 3) < -5 \times 2$

$(x – 3) < -10$

$x – 3 + 3 < -10 + 3$
$x < -7$
The solution of the given problem is $x$ has values less than $-7$. Interval notation to represent is (-infinity, $-7$)