Inequalities

Inequalities

Given two numbers $a$ and $b$, we write $a$b$) or equivalently $b>a$ ($b$ is greater than $a$) if $b-a$
is positive. Geometrically $aon the number line (see the following figure).

  • The symbol
    $a\leq b$ means either $aor $a=b$ and
  • $aand $b\leq c$.
$a

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

  1. If $a\neq b$ then $ab$.
  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 $acsides 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 $a$\dfrac{1}{a}>\dfrac{1}{b}$.
  8. If $a<0
  9. If $a\neq0$, $a^{2}>0$.

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

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