Inequality

math

Definition

Inequality

An inequality is a mathematical expression that consists of two terms, connected by a comparison symbol ($<,\le ,>,\ge$).

Example: $x<b$

Rules

Addition

Let $a,b,c$ be numbers, the following holds:

$$a<b⟺a+c<b+c$$

Multiplication by Positive Number

Let $a,b$ be numbers and $c>0$, the following holds:

$$a<b⟺a\cdot c<b\cdot c$$

Multiplication by Negative Number

Let $a,b$ be numbers and $c<0$, the following holds:

$$a<b⟺a\cdot c>b\cdot c$$

Exponentiation

Let $x,y>0$ and $r\in {\mathbb{Q}}^{+}$, then

$$x<y⟺x^r<y^r$$