Quadratic Inequalities & Modulus Functions

Quadratic inequalities and modulus functions are foundational algebraic tools that appear as sub-steps in many JEE problems across topics. Mastery of sign analysis, wavy curve method, and modulus properties is essential.

Quadratic Inequalities:

A quadratic inequality has the form $ax^{2}$ + bx + c > 0 (or >=, <, <=). To solve:

1. Find the roots of $ax^{2}$ + bx + c = 0 using the quadratic formula: x = [-b +/- $\sqrt{b^{2}-4ac}$]/(2a). 2.
   If D = $b^{2}$-4ac > 0, two real roots $\alpha$, $\beta$ ($\alpha$ < $\beta$). The quadratic can be factored as a(x-$\alpha$)(x-$\beta$).
2. For a > 0: $ax^{2}$+bx+c > 0 when x < $\alpha$ or x > $\beta$ (outside roots); $ax^{2}$+bx+c < 0 when $\alpha$ < x < $\beta$ (between roots).
3. For a < 0: signs are reversed.
4. If D = 0: one repeated root. The expression is a perfect square and never changes sign (except at the root where it equals zero).
5. If D < 0: no real roots. The expression has the same sign as 'a' for all real x.

Wavy Curve Method (Method of Intervals): For any rational inequality f(x)/g(x) > 0, factor numerator and denominator completely. Mark all zeros and undefined points on the number line. Starting from the rightmost region (where the expression is positive for leading positive coefficient), alternate signs at each simple root. For repeated roots with even multiplicity, the sign does not change; for odd multiplicity, it does.

Modulus (Absolute Value) Function: |x| = x if x >= 0, -x if x < 0. Geometrically, |x-a| represents the distance of x from a on the number line.

Key Properties:

• |x| >= 0 for all x; |x| = 0 iff x = 0
• |xy| = |x||y|; |x/y| = |x|/|y| (y != 0)
• |x+y| <= |x| + |y| (triangle inequality)
• |x-y| >= ||x| - |y|| (reverse triangle inequality)
• |x|² = $x^{2}$

Solving Modulus Equations:

• |f(x)| = a (a > 0): f(x) = a or f(x) = -a
• |f(x)| = |g(x)|: f(x) = g(x) or f(x) = -g(x), equivalently [f(x)]² = [g(x)]²
• |f(x)| = f(x) iff f(x) >= 0

Solving Modulus Inequalities:

• |f(x)| < a (a > 0): -a < f(x) < a
• |f(x)| > a (a > 0): f(x) < -a or f(x) > a
• |f(x)| <= |g(x)|: [f(x)]² <= [g(x)]², i.e., [f(x)-g(x)][f(x)+g(x)] <= 0

Quadratic in Modulus: For equations like |x|² - 5|x| + 6 = 0, substitute t = |x| (t >= 0), solve the quadratic in t, then back-substitute to find x (yielding +/- values for each valid t).

Sign of Quadratic Expression:

• If a > 0 and D < 0: $ax^{2}$+bx+c > 0 for all x in R (positive definite)
• If a < 0 and D < 0: $ax^{2}$+bx+c < 0 for all x in R (negative definite)
• Condition for f(x) > 0 for all x: a > 0 AND D < 0

The key problem-solving concept is combining sign analysis with case-based modulus removal: split the domain at points where expressions inside modulus change sign, remove the modulus with appropriate signs in each interval, and solve the resulting inequalities.