An inequality is a statement that two quantities are not necessarily equal — one is less than, greater than, or comparable to the other. A linear inequality in one variable is any statement of the form $ax + b < 0$, $ax + b \le 0$, $ax + b > 0$ or $ax + b \ge 0$, where $a \ne 0$. The strict symbols $<$ and $>$ exclude equality; the weak symbols $\le$ and $\ge$ allow it.
Solving an inequality means finding every value of the variable that makes the statement true. Unlike an equation, which usually pins the variable to one or two numbers, an inequality almost always has a whole range of solutions — an interval on the number line.
Most of the rules you already use for equations carry over unchanged. You may add or subtract the same number from both sides, and you may multiply or divide both sides by the same positive number. The one rule that catches students out is the sign-flip:
For example, $-3x \ge 6 \Rightarrow x \le -2$ — dividing by $-3$ turns $\ge$ into $\le$. The table below summarises which operations preserve the direction of the inequality and which reverse it:
| Operation on both sides | Effect on the symbol | Example |
|---|---|---|
| Add or subtract any number | unchanged | $x - 4 < 1 \Rightarrow x < 5$ |
| Multiply / divide by a positive number | unchanged | $2x \le 10 \Rightarrow x \le 5$ |
| Multiply / divide by a negative number | reversed | $-2x \le 10 \Rightarrow x \ge -5$ |
The solution is then written either in set-builder form, such as $\{x : x \le -2,\ x \in \mathbb{R}\}$, or as an interval, such as $(-\infty, -2]$. On the number line we use an open circle ($\circ$) at the endpoint for a strict inequality and a closed circle ($\bullet$) for a weak one, then shade the ray of all values that satisfy it.
When the variable is restricted to the integers ($x \in \mathbb{Z}$) or naturals ($x \in \mathbb{N}$) rather than all reals, the same algebra applies but the final answer is a discrete list, for instance $\{-2, -1, 0, 1\}$, not a continuous interval.
Deeper Insight — why the sign-flip is unavoidable: The reversal rule is not an arbitrary convention you must memorise; it is forced by what "less than" actually means on the number line. Consider the plain truth $2 < 5$. Now multiply both sides by $-1$: the numbers $-2$ and $-5$ sit on the negative side, where the order is mirrored — $-5$ lies further left than $-2$, so $-2 > -5$. Multiplying by any negative number reflects every point across zero, and reflection turns "to the left of" into "to the right of", which is exactly what flipping the symbol records. This is also why you should never multiply an inequality by an expression like $x$ whose sign you do not know — you would not know whether to flip. Keep the operations transparent: isolate the variable using additions and positive multipliers wherever possible, and reach for a negative multiplier only when you consciously remember to reverse the symbol.