Inequalities & Absolute Value

Calculus Readiness · Axiom Academy

Solve them like equations — but watch the one move that flips everything — then read the answer straight off a number line. 1. Solve It, Then Graph the Ray A linear inequality is solved with the same balanced moves as an equation: undo the constant, then undo the coefficient. Take 3x - 5 positive number, so nothing special happens — and you land on x < 4 . The solution is a whole ray, not one number 2. The One Trap: Flip the Sign Here is the move that makes inequalities different from equations. Multiplying or dividing both sides by a negative reverses their order, so you must flip the inequality sign . Watch -3x + 2 , giving x > -4 . The solution ray flips direction too. ÷ by a negative ⟹ the sign reverses Shifting both sides never changes the direction — solve freely. Scaling by a positive preserves order — the sign stays as it is. Order reverses — you must flip : , becomes . 2 -5 . Negation flips the order. Test x = 0 in the original -3x + 2 -4 ✓︎ — the flipped answer agrees, so the solution ray is correct. 3. Compound: A Band Between Two Endpoints A compound inequality traps x between two bounds at once. Solve -3 all three parts : subtract 1 to get -4 band between the endpoints — open where strict, closed where not. The left endpoint -2 : the inequality there is strict ( -2 open circle. The right endpoint 2 : here it's "or equal to" ( ), so 2 is included — a closed (filled) circle. In interval notation it gets a parenthesis: opens at -2 . It gets a bracket: closes at 2 .

This is the written version of the interactive lesson above. See the full Calculus Readiness course.