Introduction to Inequalities

Inequalities use <, >, ≤ or ≥ instead of =. They describe a range of answers rather than one.

You solve them almost exactly like equations. x + 4 < 9 becomes x < 5.

The one difference: multiplying or dividing both sides by a negative flips the sign. −2x < 6 becomes x > −3, not x < −3.

Sanity check with numbers. 3 < 5 is true. Multiply both sides by −1: −3 and −5. Since −3 is bigger, the sign must flip.

Graphing: open circle for < or >, filled circle for ≤ or ≥, then shade the direction that works.

The order axioms for ℝ state that a < b implies a + c < b + c for all c, and ac < bc only when c > 0. For c < 0 the inequality reverses. Addition never flips; multiplication flips exactly when the multiplier is negative.

Geometrically, multiplying by a negative reflects the number line through 0, and a reflection reverses order. Nothing analogous exists for equations because equality is symmetric and has no order to preserve.

Two consequences matter later. First, you may not multiply both sides by a variable expression unless you know its sign, which is why rational inequalities are solved with sign charts rather than by clearing denominators. Second, solution sets are intervals, so the natural way to write an answer is interval notation: x > −3 is (−3, ∞), with a parenthesis for strict inequality and a bracket for ≤ or ≥.