Inequality¶

Definition¶

An inequality is an order comparison between two real numbers. Strict inequalities exclude equality, while weak inequalities allow it. The notations are:

• We say $a$ is less than $b$ if $a-b$ is a negative number, denoted by $a<b\text{;}$
• We say $a$ is greater than $b$ if $a-b$ is a positive number, denoted by $a>b\text{;}$
• We say $a$ is less than or equal to $b$ if $a-b$ is a negative number or zero, denoted by $a\leq b\text{;}$
• We say $a$ is greater than or equal to $b$ if $a-b$ is a positive number or zero, denoted by $a\geq b\text{.}$

Interpretation¶

Inequalities describe the order of the real numbers: $a<b$ means that $a$ lies to the left of $b$ on the number line. The definition reduces every comparison to the sign of one number, the difference $a-b$$b$.

The figure shows the numbers that satisfy both $-1<x$ and $x\le2$$2$, written $(-1,2]$. The open circle at $-1$ marks a strict inequality, whose boundary point is excluded; the filled dot at $2$ marks a non-strict one, whose boundary point is included: $2\le2$ is true, while $2<2$ is false.

The sign of $a-b$ explains the rules in the Key Results. Adding $c$ to both sides leaves the difference unchanged, $(a+c)-(b+c)=a-b$$b$, while multiplying by $c$ turns it into $(a-b)c$$b)c$, whose sign is kept when $c>0$ and flips when $c<0$$0$; for $c=0$ both sides become $0$, so a strict inequality turns into an equality. A common slip is to multiply by a quantity of unknown sign, such as $x$ in $1/x<2$$2$; the cases $x>0$ and $x<0$ must be treated separately.

Key Results¶

• Inequalities define ordering on the real numbers.
• Inequalities are preserved under addition.
• Multiplication by negative numbers reverses inequality direction.

Examples¶

• If $a<b\text{,}$ then $a+1<b+1\text{.}$
• If $a<b$ and $c<0\text{,}$ then $ac>bc\text{.}$

Related Concepts¶

• Absolute Value
• Trichotomy of Real Numbers
• Real Number

Prerequisites¶

• Real Number

Sources¶

• Math I: Calculus · Lecture 2

Last updated: 27 October 2026 11:52