1.4 Inequalities - Avidemia

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 $a c > b c$ (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 $a c < b c$ (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 $a \neq 0$ , $a^{2} > 0$ .

The above properties remain true, if we replace $>$ by $\geq$ and $<$ by $\leq$ .