Inequality

An inequality is a relationship between two different quantities or expressions. An inequality may be expressed by a mathematical sentence that uses the following symbols:

< is less than

> is greater than

≤ is less than or equal to

≥ is greater than or equal to

≠ is not equal to

Refer to the inequality symbols page for more information on how these symbols are used.

An inequality is similar to an equation in that they both describe the relationship between two expressions. Below are some examples of inequalities:

Examples

Properties of inequalities

Inequalities, like many other relations in math, are governed by certain properties. Below are some of these properties. Note that the properties hold for the strict (< and >), as well as non-strict inequalities (≤ and ≥).

Inequality relations are converses of each other

< and > are converses. Given that

a > b

we also know that

b < a

Inequality relations are transitive

Given some values, a, b, and c, if

a < b and b < c, then a < c

It follows that if

a < b and b ≤ c, then a < c

a ≤ b and b < c, then a < c


Constants can be added to or subtracted from both sides

Like algebraic equations, we can manipulate inequalities as long as we perform operations to both sides. As long as we add some constant to the left side of an inequality, we can add that same constant to the other side of the inequality. The same is true of subtraction. Given a constant, c, if

a ≥ b, then a + c ≥ b + c

Constants can be multiplied or divided from both sides

Like addition/subtraction, multiplication and division can be applied to inequalities. It is important to note that when multiplying and dividing, we must pay attention to the sign of the constant. If a constant, c, is positive, the inequality relation remains the same. Given that c is positive, if

a ≤ b, then ac ≤ bc

a ≤ b, then ≤

However, if the constant, c, is negative, we must reverse the symbol to preserve the inequality relation. Given that c is negative, if

a ≤ b, then ac ≥ bc

a ≤ b, then ≥

The additive inverse reverses the inequality relation

If we take the additive inverse of both sides of the inequality, we must reverse the direction of the inequality symbol. Generally, given that

a ≥ b, -a ≤ -b

The multiplicative inverse reverses the inequality relation

Given that the values on either side of the equation are both positive or both negative, if we take the multiplicative inverse of both sides, we must reverse the direction of the inequality symbol. Generally

If a ≤ b,

a ≤ b, then ≥