review-of-fundamentals

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 \(<\) 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 \(ac>bc\) (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 \(\frac{1}{a}>\frac{1}{b}\) .
If \(a\neq0\) , \(a^{2}>0\) .

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

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