Inequalities

Inequalities define ordering relations between real numbers. In calculus, inequalities are extensively used to establish domain restrictions, determine function monotonicity, and evaluate limits.

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Property	Rule ( $a < b$ )	Exception / Condition
Addition	$a + c < b + c$	Valid for all real numbers c
Positive Multiplication	$a c < b c$	Valid for $c > 0$
Negative Multiplication	$a c > b c$	Reverses direction for $c < 0$
Reciprocals	$\frac{1}{a} > \frac{1}{b}$	Both a and b share the same sign

Properties of Inequalities

Suppose $a < b$ , then:

For every c: $a + c < b + c$
If c is positive, then $a c < b c$ but if c is negative, then $a c > b c$
If a and b are both positive, or are both negative, then $\frac{1}{a} > \frac{1}{b}$

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

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Properties of Inequalities

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