Inequalities

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 $a=b$ and
$a<b\leq c$ means $a<b$ and $b\leq c$.

a < b geometrically means a lies to the left of b on the number line.

The signs

lt;$ and

gt;$ 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 $\dfrac{1}{a}>\dfrac{1}{b}$.
If $a<0<b$, then $\dfrac{1}{a}<0<\dfrac{1}{b}$.
If $a\neq0$, $a^{2}>0$.

The above properties remain true, if we replace

gt;$ by $\geq$ and

lt;$ by $\leq$.

footer_h_1

footer_social_media