Inequality
Definition

An inequality is a mathematical relationship between two expressions and is represented using one of the following:

• ≤: "less than or equal to"
• <: "less than"
• ≠: "not equal to"
• >: "greater than"
• ≥: "greater than or equal to"

Inequalities involving "<", "≠" or ">" are referred to as "strict inequalities", while inequalities involving "≤" or "≥" are not.

If you "switch" the two sides of an inequality you must then reverse the direction of the inequality symbol. For example since it is true that 4 < 5, it is also true that 5 > 4.

An equation is a statement of equality between two expressions. An equation uses the equality symbol (=).

Like solutions to conditional equations, solutions to inequalities in one variable can be represented using a number line.

When considering locations along a number line, the inequality symbols can be interpreted as follows:

• ≤: "to the left of or coincidental with" or "to the left of including"
• <: "to the left of"
• ≠: "not coincidental with"
• >: "to the right of"
• ≥: "to the right of or coincidental with" or "to the right of including"

Strict inequalities are usually used when no variables are involved.

Inequalities involving a variable are sometimes referred to as "inequations".

At the present time, the term "inequality" refers to both inequations (with a variable) and simple inequalities without a variable.

Examples (Inequalities Without a Variable)

Examples (Inequalities With a Variable)

Demonstration Applet

Multiplying (or dividing) both sides of an inequality by a negative number

-2 < 2: -2 is to the left of +2 on the number line (as shown below).

Adding or subtracting either a positive or negative number to each side of the inequality will result in a true statement.

For example, if you add one to each side of the inequality (or equivalently subtract negative one from each side of the inequality), the following occurs:

If you add negative one to each side of the inequality (or equivalently subtract positive one from each side of the inequality), the following occurs:

If you multiply each side of the inequality by two, the following occurs:

IMPORTANT

If you multiply each side of the inequality by negative two, the following occurs:

Note that the result of multiplying (or dividing) both sides of a (true) inequality by a negative number is an inequality which is false, unless you reverse the direction of the inequality.

It is important to keep this in mind when solving an inequality such as -2x + 7 ≥ 25.