Course Overview: MAT 1033C Intermediate Algebra

This study guide provides a structured overview of the main topics covered in an 8-week Intermediate Algebra course (MAT 1033C). The course is organized by weekly modules, each focusing on foundational algebraic concepts essential for further study in mathematics and related fields.

Week 1: Equations, Inequalities, and Problem Solving

Linear Equations

• Definition: A linear equation is an equation of the form , where , , and are constants and is the variable.

• Solving Steps:

  1. Isolate the variable on one side of the equation.

  2. Simplify both sides as needed.

  3. Check the solution by substituting back into the original equation.

• Example: Solve .

  • Subtract 5:

  • Divide by 2:

Problem Solving and Formulas

• Application: Use algebraic equations to model and solve real-world problems, such as distance, rate, and time: .

• Formulas: Rearranging formulas to solve for a specified variable.

Linear Inequalities and Compound Inequalities

• Definition: An inequality compares two expressions using symbols such as .

• Solving: Similar to equations, but reverse the inequality sign when multiplying or dividing by a negative number.

• Compound Inequalities: Combine two inequalities, e.g., .

• Example: Solve .

  • Subtract 2:

  • Divide by -3 (reverse sign):

Week 2: Graphs and Functions

Graphing Equations and Linear Functions

• Definition: The graph of an equation is the set of all points that satisfy the equation.

• Linear Function: A function of the form .

• Graphing Steps:

  1. Find the -intercept ().

  2. Use the slope () to find another point.

  3. Draw a straight line through the points.

• Example: Graph .

Introduction to Functions

• Definition: A function is a relation where each input has exactly one output.

• Notation: denotes the value of the function at .

Slope and Equations of Lines

• Slope Formula:

• Point-Slope Form:

• Slope-Intercept Form:

Graphing Linear Inequalities

• Method: Graph the boundary line, then shade the region representing the solution set.

Week 3: Exponents, Polynomials, and Polynomial Functions; Rational Expressions

Factoring Polynomials

• Definition: Factoring is expressing a polynomial as a product of its factors.

• Special Products:

  • Difference of Squares:

  • Perfect Square Trinomial:

• Example: Factor

Rational Expressions: Multiplying, Dividing, Adding, Subtracting

• Definition: A rational expression is a fraction with polynomials in the numerator and denominator.

• Multiplying/Dividing: Multiply numerators and denominators; for division, multiply by the reciprocal.

• Adding/Subtracting: Find a common denominator before combining.

• Example:

Week 4: Rational Expressions (continued)

Simplifying Complex Fractions

• Definition: A complex fraction has a fraction in its numerator, denominator, or both.

• Method: Multiply numerator and denominator by the least common denominator (LCD) to simplify.

Dividing Polynomials: Long Division

• Process: Similar to numerical long division; divide, multiply, subtract, bring down the next term, and repeat.

• Example: Divide by .

Solving Equations with Rational Expressions

• Steps:

  1. Find the LCD of all denominators.

  2. Multiply both sides by the LCD to clear denominators.

  3. Solve the resulting equation.

  4. Check for extraneous solutions.

Week 5: Rational Exponents, Radicals, and Complex Numbers

Radicals and Radical Functions

• Definition: A radical expression contains a root, such as .

• Properties:

• Example:

Simplifying, Adding, Subtracting, and Multiplying Radical Expressions

• Simplifying: Factor out perfect squares.

• Combining Like Radicals: Only like radicals can be added or subtracted.

• Multiplying: Use distributive property and properties of radicals.

Week 6: Rational Exponents, Radicals, and Complex Numbers (continued)

Rationalizing Denominators and Numerators

• Definition: To rationalize means to eliminate radicals from the denominator or numerator.

• Method: Multiply numerator and denominator by a suitable radical.

Radical Equations and Problem Solving

• Solving: Isolate the radical, then raise both sides to the appropriate power.

• Check for extraneous solutions.

Complex Numbers

• Definition: A complex number is of the form , where .

• Operations: Add, subtract, multiply, and divide using .

• Example:

Rational Exponents

• Definition:

• Example:

Week 7: Quadratic Equations and Functions; Systems of Equations

Solving Quadratic Equations Using the Quadratic Formula

• Quadratic Equation:

• Quadratic Formula:

• Example: Solve .

  • or

Quadratic Functions and Their Graphs

• Standard Form:

• Vertex:

• Graph: Parabola opening up if , down if .

Systems of Linear Equations

• Definition: A system consists of two or more equations with the same variables.

• Solving Methods:

  • Substitution

  • Elimination

  • Graphing

• Example: Solve:

  • Add:

  • Substitute:

Week 8: Review and Final Exam

• Comprehensive review of all topics.

• Practice problems and exam strategies.

Summary Table: Weekly Topics Overview

Additional info: This guide is based on a course syllabus and weekly schedule. For detailed examples and practice problems, refer to the assigned textbook sections and recommended video playlists.