Algebra: Theory and Practice Problems

By Pimath, 21 May, 2026

1. Sets and Logic

Number Sets and Operations

Topic	Theory	Practice Problems
Number Sets: Natural, Integer, Rational, Irrational and Real Numbers Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Set Operations: Union, Intersection, Difference and Complement Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Cartesian Product Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓

Propositional Logic and Mathematical Induction

Topic	Theory	Practice Problems
Propositional Logic Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Principle of Mathematical Induction Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓

2. Numbers and Operations

2.1 Powers and Radicals

Topic	Theory	Practice Problems
Definition and Properties of Powers Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Radicals: Definition, Properties and Examples Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Operations with Radicals Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Rationalization Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓

2.2 Logarithms

Topic	Theory	Practice Problems
Definition and Properties of Logarithms Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓

3. Algebraic Expressions

Polynomials, Fractions and Absolute Value

Topic	Theory	Practice Problems
Monomials and Polynomials Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Special Products Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Polynomial Factorization Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Polynomial Division Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Ruffini's Rule Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Algebraic Fractions Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Absolute Value: Definition and Properties Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓

4. Equations

Topic	Theory	Practice Problems
Linear Equations Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Quadratic Equations Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Higher-Degree Equations Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Absolute Value Equations Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Irrational Equations / Equations with Radicals Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Exponential Equations Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Logarithmic Equations Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Parametric Equations Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓

5. Inequalities

Topic	Theory	Practice Problems
Linear Inequalities Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Quadratic Inequalities Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Higher-Degree Inequalities Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Rational Inequalities Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Absolute Value Inequalities Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Irrational Inequalities / Inequalities with Radicals Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Exponential Inequalities Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Logarithmic Inequalities Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Inequalities with Parameters Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Sign Analysis Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓

6. Systems

Systems of Equations and Inequalities

Topic	Theory	Practice Problems
Systems of Linear Equations Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Systems of Quadratic Equations Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓
Systems of Inequalities Theory ✓ Practice Problems ✓	Theory ✓	Practice Problems ✓