Given two numbers $a$ and $b$, we write $a<b$ ($a$ is less than $b$) or equivalently $b>a$ ($b$ is greater than $a$) if $b-a$ is positive. Geometrically $a<b$ means $a$ lies to the left of $b$ on the number line (see the following figure).
- The symbol $a\leq b$ means either $a<b$ or $a=b$.
- $a< b\leq c$ means $a<b$ and $b\leq c$.
Figure: $a<b$ geometrically means that $a$ lies to the left of $b$ on the number line.
The signs $<$ and $>$ are called inequality symbols and satisfy the following properties:
- If $a\neq b$ then $a<b$ or $a>b$.
- If $a>b$ and $b>c$ then $a>c$.
- If $a>b$ then $a+c>b+c$ (and $a-c>b-c$) for every $c$ (if we add a positive or negative number to both sides of an inequality, the direction of the inequality will be preserved).
- If $a>b$ and $c>d$, then $a+c>b+d$ (inequalities with the same directions can be added).
- If $a>b$ and $c>0$ then $ac>bc$ (if we multiply or divide both sides of an inequality by a positive number the direction of the inequality will be preserved).
- If $a>b$ and $c<0$ then $ac<bc$ (if we multiply or divide both sides of an inequality by a negative number, we need to reverse the inequality direction).
- If $a$ and $b$ are both positive or both negative and $a<b$ then $\frac{1}{a}>\frac{1}{b}$.
- If $a\neq0$, $a^{2}>0$.
The above properties remain true, if we replace $>$ by $\geq$ and $<$ by $\leq$.