Wavy Curve MethodJEE Main

Concept

The wavy curve (sign-scheme) method solves any factorable inequality: mark every zero and pole on the number line, then sweep from the far right where all factors are positive — the sign alternates at every simple root (and stays put at even-multiplicity roots). Read off the intervals you want.

Key formula

Derivation

Each linear factor flips sign exactly at its root. Crossing one critical point flips ONE factor → total sign flips. Even powers ((x−a)²) flip twice = no change — the 'wave' bounces.

Poles behave like zeros for sign purposes but are ALWAYS excluded from the solution; zeros are included only for ≥ / ≤.

Scenarios to explore

• Wavy Curve Method — Sign analysis of rational expressions on the number line.

Real-world applications

• Domains of √ and log expressions.
• Monotonicity: solving f'(x) > 0.
• Range problems & quadratic-in-disguise inequalities.

JEE exam tips

• Always start from the RIGHTMOST interval with + (after making every leading coefficient positive).
• ≥ vs >: square-bracket the zeros, never the poles.
• Convert x² < a² to |x| < a — pairs beautifully with the wavy sweep.

Common mistakes

• Multiplying both sides by (x−c) without knowing its sign — the cardinal sin. Move everything to one side instead.
• Including the pole in the answer.
• Forgetting even-power roots don't flip the sign.

Exam traps to avoid

• (x−1)²(x−3) > 0: the wave TOUCHES at 1 and crosses at 3 — solution excludes x = 1.
• Never cancel common factors across the inequality without tracking their sign & zeros.