Solving Inequalities

Pre-Algebra · Axiom Academy

Almost exactly like solving an equation — until you multiply or divide by a negative, and the sign flips. An equation like x = 4 has one answer. An inequality like has infinitely many — every number to the left of 4 . We picture that with a shaded ray : an endpoint, then an arrow running off in the direction of all the solutions. Open endpoint — the boundary is NOT included Closed endpoint — the boundary IS included Adding, subtracting, multiplying, or dividing by a positive number works on an inequality exactly the way it works on an equation. Do the same thing to both sides and the inequality symbol stays pointing the same way. Watch get solved by peeling 3 off both sides at once. 3. The Critical Rule: Flip on a Negative Why? Multiplying by a negative reflects every number across 0 . Start with . Reflect both across 0 and you land on -2 and -5 — but now -2 sits to the right of -5 , so the order has reversed: . The reflection literally turns the symbol around. Watch it happen, then watch get solved with the flip. Same idea with a multi-step problem — clear the constant first, then divide and flip: The symbol stays put. Solve it just like an equation. Flip the symbol. This is the only step that reverses it. You can now solve an inequality the same way you solve an equation, watch for the one step that flips the sign, and draw the answer as a shaded ray. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Pre-Algebra course.