Inequalities

Inequalities describe ranges of values rather than single solutions. They are essential in calculus for defining intervals, stating theorems, and working with absolute values. The key rules are simple but the sign-flip rule, multiply by a negative and the inequality reverses, causes errors more often than any other algebraic mistake.

Operation	Effect on $a < b$
Add any constant $c$	$a + c < b + c$
Multiply by $c > 0$	$a c < b c$
Multiply by $c < 0$	$a c > b c$ (inequality reverses)
Take reciprocals (same sign)	$\frac{1}{a} > \frac{1}{b}$

Rules for Inequalities

Suppose $a < b$ . Then the following rules hold:

Addition rule: For every $c$ : $a + c < b + c .$
Multiplication rule: If $c$ is positive, then $a c < b c,$ but if $c$ is negative, then $a c > b c .$ The inequality reverses direction when you multiply (or divide) by a negative number.
Reciprocal rule: If $a$ and $b$ are both positive, or both negative, then $\frac{1}{a} > \frac{1}{b} .$

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

Frequently Asked Questions

Why does the inequality reverse when I multiply by a negative number?

Consider

2 < 5

. Multiply both sides by

- 1

: we get

- 2

and

- 5

. On the number line,

- 2

is to the right of

- 5

, so

- 2 > - 5

. The inequality reverses. Intuitively, multiplying by

- 1

reflects both numbers across zero, which swaps their left-right order on the number line.

Does the reciprocal rule apply when

a

and

b

have opposite signs?

No. If

a < 0 < b

, then

\frac{1}{a} < 0 < \frac{1}{b}

, so the original direction

\frac{1}{a} < \frac{1}{b}

actually holds in this case, but by a different argument. The stated reciprocal rule applies specifically when both numbers are the same sign. Be careful with zero:

\frac{1}{0}

is undefined, so neither side of the inequality can be zero.

How are inequalities used in calculus?

Inequalities appear everywhere in calculus: in the epsilon-delta definition of a limit, in the squeeze theorem, in establishing bounds on errors in approximation, and in determining where functions are increasing or decreasing. The formal statement "for every

ε > 0

there exists

δ > 0

such that

| x - a | < δ

implies

| f (x) - L | < ε

" is pure inequality language.

Rules for Inequalities

Frequently Asked Questions

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