Math 98: Beginning Algebra – Course Structure and Topic Overview

Course Overview

This study guide summarizes the structure and main topics of a typical beginning algebra course, as outlined in the provided Math 98 syllabus. The course is organized into weekly modules, each focusing on foundational algebraic concepts essential for further study in mathematics.

Weekly Topic Breakdown

• Weeks 1–2: Linear Equations and Inequalities (Sections 2.1–2.7, 3.2–3.5)

• Weeks 3–4: Graphing and Introduction to Systems of Equations (Sections 3.6–3.7, 4.1–4.6)

• Weeks 5–6: Exponents, Polynomials, and Factoring (Sections 4.7–5.7, 7.1–7.2)

• Weeks 7–8: Factoring, Rational Expressions, and Equations (Sections 6.1–6.7, 7.7)

• Weeks 9–10: Rational Expressions, Roots, and Radicals (Sections 8.1–8.7, 9.1, 9.3)

• Weeks 11–12: Quadratic Equations and Review (Section 9.6, Final Exam)

Key Topics and Subtopics

Linear Equations and Inequalities

Linear equations and inequalities form the foundation of algebra. Students learn to solve for unknowns and interpret solutions in various contexts.

• Definition: A linear equation is an equation of the form $ax + b = c$, where $a$, $b$, and $c$ are constants.

• Solving Linear Equations: Isolate the variable using inverse operations.

• Linear Inequalities: Similar to equations, but solutions are ranges of values. Remember to reverse the inequality sign when multiplying or dividing by a negative number.

• Example: Solve $2x - 5 = 9$.

  • Add 5 to both sides: $2x = 14$

  • Divide by 2: $x = 7$

Solving Word Problems

Word problems require translating real-world situations into algebraic equations or inequalities.

• Key Steps:

  1. Read the problem carefully.

  2. Define variables for unknowns.

  3. Write an equation or inequality based on the relationships described.

  4. Solve and interpret the solution.

• Example: If a number increased by 8 is 15, what is the number?

  • Let $x$ be the number. $x + 8 = 15$

  • $x = 15 - 8 = 7$

Graphing

Graphing is a visual method to represent equations and inequalities, especially linear relationships.

• Coordinate Plane: The plane is defined by the x-axis (horizontal) and y-axis (vertical).

• Graphing Linear Equations: Use the slope-intercept form $y = mx + b$.

• Finding Intercepts: Set $x = 0$ for the y-intercept, $y = 0$ for the x-intercept.

• Example: Graph $y = 2x + 1$ by plotting the y-intercept (0,1) and using the slope 2 (rise 2, run 1).

Systems of Linear Equations

Systems involve solving for variables that satisfy two or more equations simultaneously.

• Methods: Substitution, elimination, and graphing.

• Example: Solve:

  • $x + y = 5$

  • $x - y = 1$

  • Add equations: $2x = 6 \Rightarrow x = 3$

  • Substitute: $3 + y = 5 \Rightarrow y = 2$

Exponents and Polynomials

Understanding exponents and polynomials is essential for manipulating algebraic expressions.

• Exponent Rules:

  • $a^m \cdot a^n = a^{m+n}$

  • $(a^m)^n = a^{mn}$

  • $a^0 = 1$ (for $a \neq 0$)

• Polynomials: Expressions with multiple terms, e.g., $3x^2 + 2x - 5$.

• Operations: Addition, subtraction, multiplication, and division (by monomials).

• Example: $(x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6$

Factoring

Factoring rewrites polynomials as products of simpler expressions, which is crucial for solving equations.

• Common Methods: Factoring out the greatest common factor (GCF), factoring trinomials, difference of squares.

• Example: $x^2 - 9 = (x + 3)(x - 3)$

Rational Expressions and Equations

Rational expressions are fractions with polynomials in the numerator and denominator.

• Simplifying: Factor and reduce common terms.

• Operations: Addition, subtraction, multiplication, and division of rational expressions.

• Solving Rational Equations: Find a common denominator, clear fractions, and solve the resulting equation.

• Example: Simplify $\frac{x^2 - 9}{x + 3} = \frac{(x + 3)(x - 3)}{x + 3} = x - 3$ (for $x \neq -3$)

Roots and Radicals

Roots and radicals involve expressions with square roots, cube roots, and higher-order roots.

• Definition: The square root of $a$ is a number $x$ such that $x^2 = a$.

• Properties:

  • $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$

  • $\sqrt{a^2} = |a|$

• Example: Simplify $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$

Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$.

• Methods of Solution: Factoring, completing the square, and the quadratic formula.

• Quadratic Formula:

  • $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: Solve $x^2 - 5x + 6 = 0$ by factoring: $(x - 2)(x - 3) = 0 \Rightarrow x = 2, 3$

Assessment Structure

• Homework: Assigned weekly, covering the current sections.

• Worksheets: Practice problems for key skills (e.g., factoring, graphing).

• Exams: Five unit exams, a midterm, and a comprehensive final exam.

• Practice Exams: Required to unlock each main exam, ensuring readiness.

Sample Weekly Schedule Table

Week	Main Topics	Assessments
1	Linear Equations and Inequalities (2.1–2.7)	Homework 1, Worksheet 1
3	Graphing, Systems of Equations (3.6–3.7, 4.1–4.6)	Exam #2, Homework 3
5	Exponents, Polynomials, Factoring (4.7–5.3, 7.1–7.2)	Exam #3, Homework 5
7	Factoring, Rational Expressions (6.1–6.6, 7.7)	Homework 7, Worksheet 3
9	Rational Expressions, Roots and Radicals (8.1–8.6)	Homework 9, Worksheet 4
11	Quadratic Equations (9.6)	Homework 11, Worksheet 5, Practice Final

Additional Info

• Some sections are skipped (e.g., 6.5, 8.3, distance formula in 8.6) as per syllabus notes.

• Students are required to show all work, especially for graphing and final review worksheets.

• Practice exams are mandatory to unlock main exams, ensuring students are prepared.