Graphical Interpretation
Every inequality has a picture, and seeing it often beats algebra. On the number line, a quadratic y = (x − 2)(x − 3) is a U-shaped parabola; the inequality < 0 simply asks where the curve dips below the x-axis, which is the interval between the roots. In two variables, a line such as 2x + 3y = 12 splits the plane into two half-planes; an inequality like 2x + 3y ≤ 12 selects one of them, and a system of such constraints carves out a feasible region whose corner points decide the maximum or minimum of any linear objective — the heart of linear programming. The graphical view also powers maxima-minima without calculus through the AM-GM inequality: for positive numbers, the arithmetic mean is at least the geometric mean, with equality only when all the numbers are equal. That single fact tells you, for instance, that x + 1/x ≥ 2 for x > 0, with the minimum 2 reached exactly at x = 1 — a classic CAT result.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Core rules & linear
| Sign-flip rule | Multiply/divide both sides by a negative ⇒ reverse the inequality |
|---|---|
| Adding a constant | a > b ⇒ a + c > b + c (direction unchanged) |
| Multiply by positive k | a > b, k > 0 ⇒ ka > kb |
| Reciprocal (same sign) | 0 < a < b ⇒ 1/a > 1/b |
| Transitivity | a > b and b > c ⇒ a > c |
CAT power-tools
| Modulus less-than | |x| < a ⇔ −a < x < a (a > 0) |
|---|---|
| Modulus greater-than | |x| > a ⇔ x < −a or x > a (a > 0) |
| Quadratic sign | a(x−p)(x−q) with a > 0: negative between roots, positive outside |
| AM-GM (n positives) | (a₁+…+aₙ)/n ≥ (a₁…aₙ)^(1/n), equality when all equal |
| AM-GM corollary | For x > 0, x + 1/x ≥ 2; x + k/x ≥ 2√k |