Linear Inequality is a mathematical condition where two mathematical expressions, which are not equal, are compared. Many symbols are used to represent linear equations, and we will discuss each of these at length further in the article.

Linear inequalities are broadly divided into two types: algebraic inequality and numerical inequality. However, in some linear equations, both of these types of Linear Inequality may occur together.

## Linear Inequalities

By definition, linear inequalities are expressions with at least one linear algebraic expression; this means a polynomial with one degree is compared with another expression of the same degree or less. And very often, these comparisons are between two unequal expressions.

## Symbols Used In Linear Inequalities

The symbols, which are used to represent linear inequalities, are as follows.

### Not Equal

A not equal sign indicates that two values in a linear equation are not of the same value.

E.g., Y ≠ 67

### Less Than

As the name suggests, a less than sign tells that one of the two values in a linear equation is less in magnitude than the other.

E.g – z 78 < √5

### Greater Than

This symbol is used in conditions where you need to indicate that one of the two values in the expression is more than the other value.

E.g – 18 30z > 28 86y

### Greater Than Or Equal To

When one of the two expressions is greater or is almost greater than the other, we can use a greater than equal symbol to equate the two values in linear inequalities.

E.g – -67 – √47z ≥ 19

### Less Than Or Equal To

This expression is the opposite of the greater than or equal. This is used to indicate that one of the two values in the linear equation is small or equal to the other value.

E.g., Z ≤ 56

In both greater than or equal to and less than or equal to, a variable is present in the equation, and it’s the value of the variable that affects the value of either of the two values in the expression.

## Rules Of Inequality

There are generally four types of operations in linear inequality. They are addition, subtraction, multiplication and division. Let’s discuss these rules before moving further.

- Additional Rule

In the additional rule of linear inequality, you have to add the same number to both sides of the equation. This makes the equation an equivalent inequality. And also, there is no requirement to change the signs while adding.

For example: if x > y and x < y,

When we add 4 to both the sides, it becomes

= x + 4 > y + 4 & x + 4 < y + 4, respectively.

- Subtraction rule

When we subtract any number from both sides in the equation of linear inequality, it becomes the equivalent inequality. In this, the operation symbol does not change either.

For example, if x > y and x < y,

If you subtract 5 from both sides of the equation, you will get

= x – 5 > y – 5 and x – 5 < y – 6

- Multiplication Rule

On multiplying both sides of linear inequality by the same number, there is no change in symbol required in the equation. And makes the equation an equivalent one.

For example :

If x < y and b < 0

When you multiply both sides of the equation by b, you will obtain

= x * b > y * b

And if x > y and b < 0

Then, on multiplying, you will get,

= x * b < y * b

- Division Rule

This rule implies that after dividing both the sides of a linear inequality with a positive value, you will get an equation in which the inequality still exists, and the symbols of inequality remain unchanged.

For example :

If x < y and b < 0

When you divide both sides of the equation by b, you will obtain

= x / b > y / b

And if x > y and b < 0

Then, on dividing them you will get,

= x / b < y / b

However, when the equation is divided by a negative value, the symbol of inequality gets reversed, changing the relation between the terms in the equation.

## Graphical Representation Of Linear Inequalities

To represent linear inequality on a graph, we need to follow certain rules, which are as follows –

- When there are one or more variables.

- When the inequality has an ≥ or ≤ symbol, we need to draw a thick line to show that all the coordinates on the drawn lines are in the solution set of the equation.
- Whereas, when the inequality has > or < symbol, we need to draw a dotted line to show that all the coordinates on the drawn lines are out of the solution set of the equation.

- When the linear inequality is expressed as px qy > r, px by ≥ r, px qy < r or px qy ≤ r (p ≠ 0, q ≠ 0)

## Some Important Results Of Linear Inequality

Some results of inequality that you must remember while plotting the graphical representation of an equation. These results are as follows –

- If x, y ∈ R and y ≠ 0, then

- xy > 0 or x y > 0 ⇒ x and y has the same signs
- xy < 0 or x y < 0 ⇒ x and y has opposite signs

- When x is a positive real number i.e., x > 0, then

- | a | < x ⇔ – x < a < x

| a | ≤ x ⇔ – x ≤ a ≤ x

- | a | > x ⇔ a < – x or a > x

| a | ≥ x ⇔ a ≤ – x or a ≥ x

## Key Takeaways

A linear inequality is an expression, which contains two values that are not equal. Hence, their relationship is represented using different symbols such as equal to, greater than, less than, greater than equal to or less than equal to. The relation can well be indicated by simply not being equal to.

The operations you perform on these linear equations must satisfy the rules of addition, subtraction, multiplication and division depending on the equation you wish to perform and the type of value (positive or negative) you are using to perform that operation.

Moreover, while you are representing such equations on a graph, you need to ensure that it follows the rule of graphical representation. Whenever you plot a graph, you require the results mentioned above, so you must remember these results and their conditions.