An inequality compares two quantities without requiring them to be equal. Symbols such as $<$, $>$, $\le$ and $\ge$ describe which values are smaller or larger.

A linear inequality usually has many solutions rather than a single one. For example,

$$x+2<5$$

is true for every $x<3$.

Transforming inequalities

Most operations used for equations work in the same way for inequalities: adding or subtracting the same quantity from both sides preserves the comparison, as does multiplying or dividing by a positive number.

Multiplying or dividing by a negative number reverses the inequality sign. For example,

$$-2x<6$$

becomes

$$x>-3.$$

The reversal is necessary because multiplying by a negative number reverses order: although $2<5$, we have $-2>-5$.

Solution sets

The solutions of an inequality form a set of values. The solution $x>-3$ can be written with interval notation as

$$(-3,\infty).$$

Compound inequalities combine conditions. For example,

$$1\le x<4$$

means that both conditions hold at once and corresponds to the interval

$$[1,4).$$