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

\(ageometrically means \(a\) lies to the left of \(b\) on the number line.

The signs \(<\) and \(>\) are called inequality symbols and satisfy the following properties:

  1. If \(a\neq b\) then \(aor \(a>b\) .
  2. If \(a>b\) and \(b>c\) then \(a>c\) .
  3. 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).
  4. If \(a>b\) and \(c>d\) , then \(a+c>b+d\) (inequalities with the same directions can be added).
  5. 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).
  6. If \(a>b\) and \(c<0\) then \(ac(if we multiply or divide both sides of an inequality by a negative number, we need to reverse the inequality direction).
  7. If \(a\) and \(b\) are both positive or both negative and \(athen \(\frac{1}{a}>\frac{1}{b}\) .
  8. If \(a\neq0\) , \(a^{2}>0\) .

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