Module 1: Foundations of Algebra

Lesson 1: Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable has an exponent of one. Solving these equations is a fundamental skill in algebra.

• Definition: A linear equation in one variable has the form , where and are constants.

• Key Steps: Isolate the variable by performing inverse operations.

• Example: Solve . Subtract 3: . Divide by 2: .

Lesson 2: Applications of Linear Equations

Linear equations are used to model and solve real-world problems involving constant rates or relationships.

• Application: Distance, rate, and time problems; financial calculations.

• Example: If a car travels at 60 mph for hours, the distance is .

Lesson 3: Solving Linear Inequalities in One Variable

Linear inequalities express a range of possible values for a variable.

• Definition: An inequality such as .

• Key Steps: Solve similarly to equations, but reverse the inequality sign when multiplying/dividing by a negative.

• Example: Solve . Add 5: . Divide by 3: .

Lesson 4: Unions, Intersections, and Compound Inequalities

Compound inequalities combine two or more inequalities using "and" (intersection) or "or" (union).

• Intersection (and): Solutions satisfy both inequalities.

• Union (or): Solutions satisfy at least one inequality.

• Example: (intersection); or (union).

Lesson 5: Absolute Value Equations and Inequalities

Absolute value equations involve the distance from zero, always non-negative.

• Definition: means or .

• Example: Solve . or ; or .

Module 2: Linear Equations and Systems

Lesson 6: Linear Equations

Linear equations in two variables are represented as and graph as straight lines.

• Graphing: Plot by finding intercepts or using slope-intercept form .

• Example: ; .

Lesson 7: Linear Inequalities in Two Variables

Linear inequalities in two variables define regions in the coordinate plane.

• Graphing: Use a dashed or solid line for the boundary; shade the solution region.

• Example: .

Lesson 8: Systems of Linear Equations in Two Variables

Systems of equations involve finding values that satisfy all equations simultaneously.

• Methods: Graphing, substitution, elimination.

• Example: Solve and .

Lesson 9: Applications of Systems of Equations

Systems model real-world scenarios with multiple constraints.

• Application: Mixture problems, cost analysis, supply and demand.

• Example: Find the price of two items given total cost and difference.

Module 3: Polynomials and Factoring

Lesson 10: Arithmetic Operations with Polynomials

Polynomials are expressions with terms of the form . Operations include addition, subtraction, and multiplication.

• Example: .

Lesson 11: Functions and Polynomial Notation

Polynomials can be represented as functions, .

• Notation: indicates the output for input .

Lesson 12: Operations on Polynomial Functions

Polynomial functions can be added, subtracted, multiplied, and composed.

• Example: , , .

Lesson 13: Factoring Polynomials and Special Factoring

Factoring expresses a polynomial as a product of simpler polynomials.

• Methods: Greatest common factor, difference of squares, trinomials.

• Example: .

Lesson 14: Solving Polynomial Equations by Factoring

Set the polynomial equal to zero and factor to find solutions.

• Example: ; ; or .

Module 4: Rational Expressions and Quadratics

Lesson 15: Operations on Rational Expressions

Rational expressions are quotients of polynomials. Operations include addition, subtraction, multiplication, and division.

• Example: .

Lesson 16: Solving Rational Equations

Rational equations are solved by finding a common denominator and solving the resulting equation.

• Example: ; ; ; .

Module 5: Radicals and Quadratic Equations

Lesson 19: Simplifying and Operations on Radicals

Radicals involve roots, such as square roots. Simplifying involves expressing in simplest form.

• Example: .

Lesson 20: Complex Numbers

Complex numbers have the form , where .

• Example: .

Lesson 21: Solving Quadratic Equations

Quadratic equations have the form .

• Methods: Factoring, completing the square, quadratic formula.

• Quadratic Formula:

Lesson 22: Equations Quadratic in Form, Applications, and Formulas

Some equations can be rewritten as quadratics for solving.

• Example: (let ).

Lesson 23: Quadratic Analysis with Applications

Quadratics model projectile motion, area problems, and optimization.

• Application: Maximum/minimum values using vertex formula .

Module 6: Exponential and Logarithmic Functions

Lesson 24: Exponential Functions and Equations

Exponential functions have the form .

• Properties: Rapid growth or decay; , .

• Example: .

Lesson 25: Logarithmic Functions and Equations

Logarithms are the inverse of exponentials: means .

• Properties: .

• Example: because .

Lesson 26: Solving Logarithmic Equations

Logarithmic equations are solved by rewriting in exponential form or using properties.

• Example: ; .

Key Dates and Exam Information

• Midterm Exam: Covers Modules 1-3. Review all lessons and practice problems.

• Final Exam: Comprehensive, covers all modules. Review all lessons, formulas, and applications.

• Important Dates: Last day to drop/add, holidays, withdrawal deadlines, and grade due dates are listed for student planning.

Additional info: The above notes are structured by module and lesson, expanding on the brief assignment outline to provide definitions, examples, and key formulas for each major topic in College Algebra.