Inequalities

An inequality is solved almost exactly like an equation, using the same inverse operations.

The one difference

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

Why it reverses

3 < 5 is true. Multiplying both sides by −1 gives −3 and −5, and −3 is greater than −5. The order genuinely flips.

Graphing

An open circle excludes the endpoint, for < and >. A closed circle includes it, for ≤ and ≥.

Compound inequalities

An "and" statement needs both parts true, giving an overlap. An "or" statement needs either, giving two regions.

Many solutions

An inequality usually has infinitely many solutions, which is why the answer is a region rather than a number.

One rule behaves differently

Inequalities solve exactly like equations except that multiplying or dividing both sides by a negative reverses the symbol. Every other step is identical, which makes the exception easy to forget.

Why the reversal happens

3 < 5, but multiplying both by −1 gives −3 > −5. Multiplying by a negative reflects both numbers across zero, which reverses their order. The rule follows from that reflection.

The answer is a set of values

x > 4 describes infinitely many numbers. It can be written as an inequality, drawn on a number line, or given in interval notation, and different contexts prefer different forms.

Test a value to confirm the direction

Pick a number from your solution range and check it satisfies the original inequality. That single test catches a forgotten reversal immediately, which is the only real hazard in the topic.