TMUA Inequalities

The single most common error on TMUA inequalities is treating them like equations. You may divide both sides by a positive quantity freely; you may not divide both sides by an expression whose sign you do not know. Every hard inequality on the paper is built around that distinction.

• Quadratic inequalities. Sign of a quadratic between and outside its roots.
• Rational inequalities. A quotient is non-negative exactly where its numerator and denominator share a sign — sign-line analysis, not algebraic clearing of denominators.
• Modulus inequalities. $|x - a| < b$ unpacks to $a - b < x < a + b$; modulus on both sides handled by squaring when both sides are non-negative.
• Inequality-derived parameter conditions. “For all $x$, $f(x) > 0$” becomes “the discriminant is negative and the leading coefficient positive.”

The move. When the inequality involves a quotient, do not clear the denominator. Build the sign line of each factor, then read off which regions have a non-negative quotient.