In mathematics, an inequality is a relation between two values when they are different. Inequality means the condition of being unequal or lack of equality. Like x < y, x > y, x $\leq$ y etc. The sign ">" means is greater than, sign "<" means is less than. In each case the sign opens towards the larger number.

For Example: 3 < 6 ("3 is less than 6") and 6 > 3 ("6 is greater than 3"). The inequalities can be used for algebraic expressions also. For example: x + 9 < 25 will have all such values of x as solution for whom the the value of the expression (x + 9) will be less than 25.
There are two more symbols in inequality: less than or equal to ($\leq$) and greater than or equal to ($\geq$). If it is given that $x\geq 8$ it will imply that the value of x will be greater than or equal to 8, whereas x > 8 means that the value of x cannot be 8.

## What are Inequalities?

In general, an inequality is a statement about the relative size or order of two objects or about whether they are the same or not. Inequality is actually an equation, we can solve it just like equations but it has no exact answer.

We have two basic senses of inequalities as < and >.

Open circle or round parenthesis denotes the number is not included.

Closed circle or square parenthesis denotes the number is included.

Lets start with a one dimensional graph. It will contain only a one axis, say an x-axis. Here, we can consider two types of inequality:

1) Simple Inequality
2) Complex Inequality

### Simple Inequality:

This expression will contain only one inequality.

### Examples on Simple Inequality

Given below are some examples on simple inequality.

Example 1:

x < 7
According to this inequality, x will take all values less than 7. Also, as there is no equal sign, it means 7 is not included, so we use the open circle.

Example 2:

x $\geq$ -10
According to this inequality, x will take all the values greater than or equal to -10. Since there is an equal sign, -10 will be included, so we use a closed circle.

### Complex Inequality:

This expression will contain two inequalities. That is two conditions will be combined together.

### Examples on Complex Inequality

Given below are some examples on Complex Inequality.

Example 1:

2 $\leq$ x < 6

This means x will take values between 2 and 6. Here, 2 must be included, where as 6 will be excluded.

## Solving Inequalities

A solution to any inequality is any number that makes the inequality true. Inequality contains all characteristics of an equation except that the equality sign changes to inequality. Inequality will contain many solutions.

The follow steps have to be followed:
• Add or subtract so that the coefficient and variable are on either side of the equal sign, from both sides.
• Divide or multiply the coefficients of the variable, to isolate the variable.
• Swapping left and right hand sides will change the direction of inequality.
Any of the steps can be followed in any order, according to the question provided.

To solve an inequality we apply two principles:

a) Addition Principle for Inequality: If x > y then x + z > y + z.

b) Multiplication Principle for Inequality:

If x > y and z is any positive number then xz > yz and

if x > y and z is negative then xz < yz (the sign is reversed).

## Graphing Inequalities

Graphing inequalities on number lines is done to represent the solution to inequalities visually. A solution to an inequality is the value that satisfies the inequality. The concept of inequality can be understood by using the number line system.

### Graphing Inequalities on a Number Line:

Graphing inequalities on number lines is similar to graphing numbers. A number line is a horizontal line that has points which correspond to number. The point are placed according to the value of the number they correspond to. A number line containing the whole numbers and integers and the points are equally spaced.

Following two things need to be remembered, when we draw a graph on number line:

1) If we have "greater then or equal to" or "less than and equal to" sign in an inequality then we mark
a circle that is filled in is placed on the number line to show that the number denoted at the circle
is included in the solution.

2) If we have "greater then" or "less than" sign in an inequality then we mark an open circle is placed
on the number line to show that the number denoted at the circle is not included in the solution.

Example 1: Graph of x $\leq$ 7.
Solution: Here x $\leq$ 7, so we use a closed circle hat is filled to indicate that 7 is included.

Example 2: Graph of x > -3.
Solution: Here x > -3, so this time we can mark an open circle to indicate that -3 is not included.

Example 3: Solve continued inequality -4 < x < 2.
Solution: In the above inequality, we see that x > -4 and x < 2, so we use an open circle to indicate
that -4 is not included as well as 2 is not included.

Example 4: Graph -3 < x $\leq$ 5.
Solution: The problem is a complex inequality problem. We need to show a graph of all numbers
between -3 and 5. Here -3 is not included in the solution while 5 is.

• When the inequality is divided or multiplied by (-1), the inequality sign will get reversed.
• The steps needed to be taken so that direction of the inequality is not effected are:
a) Add or subtract same number on both sides.
b) Multiply or divide by positive number on both the sides.

## Inequalities Examples

Given below are some examples on solving inequality.

Example 1: Solve x + 4 > 7

Solution:
We have x + 4 > 7

Using the addition principle for inequality, add -4 to both sides of the given inequality,

x + 4 -4 > 7 - 4

or x > 3.

Example 2: Solve 2x + 5 < 9

Solution:

The first priority is to isolate x. For that, subtract 5 from both sides,
2x + 5 - 5 < 9 - 5

or 2x < 4

Divide both sides by 2

$\frac{2x}{2}$ < $\frac{4}{2}$

So, the solution is x < 2

Example 3: Solve -4x $\leq$ 3x - 14

Solution:

To solve this, we need to isolate x. Subtract 3x from both the sides

-4x - 3x $\leq$ 3x - 14 - 3x

or -7x $\leq$ -14

Divide by 7 on both the sides

$\frac{-7x}{7}$ $\leq$ $\frac{-14}{7}$

or -x $\leq$ -2

Multiply -1 on both sides. This will reverse the inequality sign.

-x (-1) $\geq$ -2(-1)

or x $\geq$ 2.

Example 4:
Solve -2x < 4

Solution: Given that -2x < 4

Using multiplication property, multiply both sides of the inequality by -0.5, this reverse the sign,

(-0.5)(-2x) > (0.5) (4)

or x > -2