Order properties of the real numbers

The rules that keep or flip an inequality

For real numbers with $a < b$:

Operation	Result	Condition
add or subtract any $c$	$a + c < b + c$	always
multiply or divide by $c$	$ac < bc$	$c > 0$
multiply or divide by $c$	$ac > bc$ (reversed)	$c < 0$
take reciprocals	$\frac{1}{a} > \frac{1}{b}$ (reversed)	$a$ and $b$ have the same sign
square	$a^2 < b^2$	$0 \leq a < b$
take an odd power or odd root	$a^3 < b^3$, $\sqrt[3]{a} < \sqrt[3]{b}$	always

Odd powers and odd roots are increasing on all reals, so they always preserve order. Even powers preserve order only between non-negative numbers: $-3 < 2$ but $9 > 4$. Reciprocals reverse order within one sign, but if $a < 0 < b$ then $\frac{1}{a} < 0 < \frac{1}{b}$ and the order stays as it was.

Two inequalities in the same direction can be added: from $a < b$ and $c < d$, $a + c < b + d$. They cannot in general be subtracted or divided, and they can be multiplied only when every term is positive.

Powers and roots of numbers between 0 and 1

For $x > 1$, raising to a higher power makes the number larger: $\sqrt{x} < x < x^2 < x^3$. For $0 < x < 1$ the order turns round, because multiplying by a number less than 1 makes things smaller:

$$0 < x^3 < x^2 < x < \sqrt{x} < 1 < \frac{1}{x}.$$

Test with $x = \frac{1}{4}$: $x^2 = \frac{1}{16}$, $\sqrt{x} = \frac{1}{2}$, $\frac{1}{x} = 4$. For $-1 < x < 0$ the signs matter too: odd powers stay negative, even powers become positive, and $\frac{1}{x} < -1$.

Comparing fractions

To compare $\frac{a}{b}$ and $\frac{c}{d}$ with positive denominators, cross-multiply: $\frac{a}{b} < \frac{c}{d}$ exactly when $ad < bc$. This is just multiplying both sides by the positive number $bd$. If a denominator could be negative, cross-multiplying may need a reversal, so check signs first.

Other quick comparisons: compare each fraction with a benchmark such as $\frac{1}{2}$ or 1, or compare how far each falls short of 1. For instance $\frac{12}{13}$ is $\frac{1}{13}$ below 1 and $\frac{14}{15}$ is $\frac{1}{15}$ below 1, so $\frac{14}{15}$ is larger.

Comparing roots

Two positive numbers compare in the same way as their squares, cubes or any positive power. So to compare roots, raise both to a power that clears every root:

• $\sqrt[3]{3}$ against $\sqrt{2}$: raise to the 6th power, giving $9$ against $8$, so $\sqrt[3]{3} > \sqrt{2}$.
• $\sqrt{2} + \sqrt{3}$ against $\sqrt{10}$: both are positive; square them. $(\sqrt{2} + \sqrt{3})^2 = 5 + 2\sqrt{6}$ and $(\sqrt{10})^2 = 10$, so compare $2\sqrt{6}$ with $5$. Square again: $24 < 25$. Hence $\sqrt{2} + \sqrt{3} < \sqrt{10}$.

Squaring is safe here only because both sides are positive. Before squaring, always confirm the sign of each side.

"Must be true" questions

A typical question gives an inequality between letters and asks which statement must follow. A statement that must be true has to hold for every allowed value, including negatives, zero (if allowed) and fractions. The fastest test is to look for a counterexample: try a negative number, a number between 0 and 1, and numbers of opposite signs. One failure removes an option; the survivor should then be justified by a rule from the table, not by the examples.

Intervals

Solution sets are written as intervals: $(a, b)$ excludes both ends, $[a, b]$ includes them, and $\infty$ always takes a round bracket. Between any two different real numbers there is another (their average), so an open interval like $(0, 1)$ has no smallest or largest element.