Introduction to Inequalities

Last Updated : 28 Apr, 2026

An inequality is a mathematical statement that compares two quantities or expressions and shows their relative size using symbols such as >, <, ≥, ≤, or ≠ instead of stating that they are equal.

inequality_symbols

The image above shows the basic symbols used in inequalities.

Inequality Symbols

Inequality symbols are listed below:

Inequality Name

Symbol

Expression

Description

Greater than 

>

x > a

x is greater than a

Less than

<

x < a

x is less than a

Greater than or equal to 

≥

x ≥ a

x is greater than or equal to a

Less than or equal to

≤

x ≤ a

x is less than or equal to a

Not equal 

≠

x ≠ a

x is not equal to a

Rules of Inequalities

There are various rules in inequalities to help us relate to and solve various inequalities. Some of these rules are discussed as follows:

Rule 1:

If a, b, and c are three numbers, then the inequality between these numbers follows the transitive property.

  • If a > b and b > c, then a > c
  • If a < b and b < c, then a < c
  • If a ≥ b and b ≥ c, then a ≥ c
  • If a ≤ b and b ≤ c, then a ≤ c

Rule 2:

If the LHS and RHS of the expressions are exchanged, then the inequality reverses. It is called the converse property.

  • If a > b, then b < a
  • If a < b, then b > a
  • If a ≥ b, then b ≤ a
  • If a ≤ b, then b ≥ a

Rule 3:

If the same constant k is added or subtracted from both sides of the inequality, then both sides of the inequality are equal.

  • If a > b, then a + k > b + k
  • If a > b, then a - k > b - k

Similarly, for other inequalities.

  • If a < b, then a + k < b + k
  • If a < b, then a - k < b - k
  • If a ≤ b, then a + k ≤ b + k
  • If a ≤ b, then a - k ≤ b - k
  • If a ≥ b, then a + k ≥ b + k
  • If a ≥ b, then a - k ≥ b - k

The direction of the inequality does not change after adding or subtracting a constant.

Rule 4:

If k is a positive constant that is multiplied or divided by both sides of the inequality, then there is no change in the direction of the inequality.

  • If a > b, then ak > bk
  • If a < b, then ak < bk
  • If a ≤ b, then ak ≤ bk
  • If a ≥ b, then ak ≥ bk

If k is a negative constant that is multiplied or divided by both sides of the inequality, then the direction of the inequality gets reversed.

  • If a > b, then ak < bk
  • If a > b, then ak < bk
  • If a ≥ b, then ak ≤ bk
  • If a ≤ b, then ak ≥ bk

Rule 5:

The square of any number is always greater than or equal to zero.

  • a2 ≥ 0

Rule 6:

Taking square roots on both sides of the inequality does not change the direction of the inequality.

  • If a > b, then √a > √b
  • If a < b, then √a < √b
  • If a ≥ b, then √a ≥ √b
  • If a ≤ b, then √a ≤ √b

Interval Notation for Inequalities

Important points for writing intervals for inequalities:

  • In case of greater than or equal to (≥) or less than or equal to (≤), the end values are included, so closed or square brackets [ ] are used.
  • In case of greater than (>) or less than (<), the end values are excluded, so open brackets () are used.
  • For both positive and negative infinity, open brackets () are used.

The following table represents intervals for different inequalities:

Inequality

Interval

x > a

(a, ∞)

x < a

(-∞, a)

x ≥ a

[a, ∞)

x ≤ a

(-∞, a]

a < x ≤ b

(a, b]

Note: There are various types of inequalities based on the degree or other aspects such as:

Graph for Inequalities

Inequalities are either with one variable or tw,o or we have a system of inequalities, all of which can be graphed in the Cartesian plane if it only contains two variables. Inequalities in one variable are plotted on real lines, and two variables are plotted on the Cartesian plane. 

Graph for Linear Inequalities with One Variable

From the following table, we can understand how to plot various Linear Inequalities with One Variable on a real line.

Inequality

Interval

Graph

x > 1

(1, ∞)

Graph for Linear Inequality x > 1
Linear Inequalities with One Variable

x < 1

(-∞, 1)

x ≥ 1

[1, ∞)

Graph for Linear Inequality x ≥ 1

x ≤ 1

(-∞, 1]

Graph for Linear Inequality x <= 1

Graph for Linear Inequalities with Two Variables

Let's take an example of linear inequalities with two variables.

Consider the linear inequality 20x + 10y ≤ 60, as the possible solutions for given inequality are (0, 0), (0,1), (0, 2), (0,3), (0,4), (0,5), (0,6), (1,0), (1,1), (1,2), (1,3), (1,4), (2,0), (2,1), (2,2), (3,0), and also all the points beyond these points are also the solution of the inequality.

Let's plot the graph from the given solutions.

Graph for Linear Inequalities with Two Variable

The shaded region in the graph represents the possible solutions for the given inequality.

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