Mathematical Inequalities for JEE Main: Deep-Dive

Inequalities appear across algebra, calculus, and trigonometry in JEE Main — sometimes as a direct problem and often as a sub-step in optimisation. The AM-GM inequality, the sign-scheme method for rational inequalities, and modulus inequalities are the three highest-frequency tools. Building a clear mental menu of which tool fits which structure is the key to fast, confident inequality work.

The AM-GM Inequality

For positive numbers a and b: (a + b)/2 ≥ √(ab), with equality iff a = b. For three positive numbers: (a + b + c)/3 ≥ ∛(abc). The general form: AM ≥ GM ≥ HM, with equality iff all numbers are equal. The primary JEE use: find the minimum of expressions like x + 1/x (minimum = 2, at x = 1), or ax + b/x (minimum = 2√(ab), at x = √(b/a)). The equality condition is always checked — it tells you where the minimum is achieved. For maximisation, apply AM-GM after rewriting the expression. Connects to our algebraic identities guide for the symmetric function connection.

The Sign-Scheme (Wavy Curve) Method

For inequalities like (x−1)(x−3)/(x−2) > 0: find all critical values (roots and points where the expression is undefined): x = 1, 2, 3. Plot on a number line. The expression is positive/negative in alternating intervals, starting from the rightmost interval (where all factors are positive). Moving left, sign flips at each critical point (for odd-power factors). At even-power critical points, the sign does not flip. The solution is the union of intervals where the expression has the required sign. This wavy-curve method is the systematic approach to all factored polynomial/rational inequalities.

Quadratic Inequalities and the Discriminant

For ax² + bx + c < 0 with a > 0: if discriminant D = b² − 4ac < 0, the quadratic is always positive (no solution for < 0). If D ≥ 0 with roots α, β (α < β): the quadratic is negative for α < x < β. If a < 0 and D < 0: the quadratic is always negative. The sign diagram for a quadratic opens upward (a > 0) — the parabola is below the x-axis between roots. This geometric picture prevents sign errors. For the root analysis connection see our quadratic equations guide.

Modulus Inequalities

|x − a| < r means −r < x − a < r, i.e. a − r < x < a + r (interval around a). |x − a| > r means x < a − r or x > a + r (exterior of interval). The triangle inequality: |a + b| ≤ |a| + |b|, with equality iff a and b have the same sign. The reverse: ||a| − |b|| ≤ |a − b|. These modulus inequalities appear in complex number magnitude bounds and in bounding sums of terms. Always square both sides when both sides are non-negative — this is often the fastest resolution of a modulus inequality.

Cauchy-Schwarz Inequality

(a₁b₁ + a₂b₂ + ... + aₙbₙ)² ≤ (a₁² + ... + aₙ²)(b₁² + ... + bₙ²), with equality iff a₁/b₁ = a₂/b₂ = ... = aₙ/bₙ. The two-variable case (a₁b₁ + a₂b₂)² ≤ (a₁²+a₂²)(b₁²+b₂²) is the most JEE-testable. It gives the maximum of a dot product for fixed magnitudes, and can be used to bound expressions like (x+y)² ≤ 2(x²+y²). Recognise a Cauchy-Schwarz structure when you see a product of sums of squares bounding a square of a sum. After mastering all inequality tools, take a free mock test on algebra.