Inequalities

Given two numbers and , we write ( is less than ) or equivalently ( is greater than ) if is positive. Geometrically means lies to the left of on the number line (see the following figure).

The symbol means either or and
means and .

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

The signs and are called inequality symbols and satisfy the following properties:

If  then  or .
If  and  then .
If  then  (and ) for every  (if we add a positive or negative number to both sides of an inequality, the direction of the inequality will be preserved).
If  and , then  (Inequalities with the same directions can be added).
If  and  then  (If we multiply or divide both sides of an inequality by a positive number, the direction of the inequality will be preserved).
If  and  then  (If we multiply or divide both sides of an inequality by a negative number, we need to reverse the inequality direction).
If  and  are both positive or both negative and  then .
If , then .
If , .

The above properties remain true, if we replace by and by .

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