A checklist for the oral exam

We collect in a single table the correspondence between the type of equation or inequality and the resolving method. It is the mental map to keep in mind at the oral exam: once the type is recognised, the method almost follows by itself.

In brief — Type → method

Type	Method
Quadratic equation	Quadratic formula
Rational equation	EC, least common multiple, reduce to the numerator
Rational/product inequality	Sign analysis, table
Irrational equation	Isolate the root, EC, sign-agreement condition, raising to a power
Irrational inequality Type I ($\sqrt{A}\le B$)	System of 3 conditions
Irrational inequality Type II ($\sqrt{A}\ge B$)	Union of 2 cases
Equation with absolute value	Split the cases $A\ge 0$, $A<0$
Inequality with absolute value	Union of systems
Exponential equation	Same base, or substitution, or logarithm
Exponential inequality	Same base + monotonicity (direction!)
Logarithmic equation	Combine logarithms, exploit injectivity, EC
Logarithmic inequality	Monotonicity (direction!) + EC

Links

Topics: Unifying methods
Concepts: Exponential inequality · Irrational inequality · Logarithmic inequality · Quadratic equation · Exponential equation · Irrational equation · Logarithmic equation · Substitution · Sign analysis · Absolute value
Skills: Reasoning by cases · Solving inequalities · Solving equations