Solving Inequalities

An inequality is solved almost exactly like an equation: undo, both sides, same move.

One rule is different

Multiplying or dividing both sides by a negative reverses the inequality sign.

Why it reverses

3 < 5 is true. Multiply both by −1 and you get −3 and −5; on the line −3 is now the larger.

The answer is a range

x > 4 has infinitely many solutions. A closed circle includes the endpoint; an open one does not.

One rule differs from equations

Inequalities are solved exactly like equations, except that multiplying or dividing both sides by a negative number reverses the direction of the symbol. Every other step is identical.

Why the flip happens

3 < 5 is true, but multiplying both by −1 gives −3 and −5, and −3 > −5. Multiplying by a negative reflects both numbers across zero, reversing their order. The rule is a consequence of that reflection.

The answer is a set, not a number

x > 4 describes infinitely many values. Express it as an inequality, on a number line, or in interval notation — (4, ∞). All three say the same thing, and different contexts prefer different forms.

Test a value from your solution

If you solve to x < −3, try x = −4 in the original inequality. If it holds, your direction is right. Testing one value inside the range catches a forgotten flip immediately.