Course Overview: Intermediate Algebra

This study guide summarizes the main topics and subtopics covered in an Intermediate Algebra college course, based on the assignment list provided. The course covers foundational concepts in algebra, including equations, inequalities, functions, systems, polynomials, factoring, and rational expressions. Each topic is essential for building mathematical reasoning and problem-solving skills.

Linear Equations and Inequalities

One Variable Linear Equations & Word Problems

Linear equations in one variable are fundamental to algebra and are used to model real-world situations.

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

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

• Word Problems: Translate real-life scenarios into equations and solve for the unknown.

• Example: Solve .

Proportions & Clearing Fractions

Proportions compare two ratios, and clearing fractions simplifies equations for easier solving.

• Proportion: An equation stating two ratios are equal: .

• Clearing Fractions: Multiply both sides by the least common denominator (LCD) to eliminate fractions.

• Example: Solve by cross-multiplying.

Literal Equations & D=RT

Literal equations involve solving for one variable in terms of others. The formula relates distance, rate, and time.

• Literal Equation: An equation with multiple variables, e.g., .

• Solving for a Variable: Rearrange the equation to isolate the desired variable.

• Distance Formula:

• Example: Solve for : .

Inequalities & Compound Inequalities

Inequalities express relationships where values are not necessarily equal. Compound inequalities combine two inequalities.

• Linear Inequality: or

• Compound Inequality: or or

• Solving: Use similar steps as equations, but reverse the inequality sign when multiplying/dividing by a negative.

• Example: Solve .

Absolute Value Equations & Inequalities

Absolute value measures the distance from zero. Equations and inequalities involving absolute value require special consideration.

• Definition: is the distance of from zero.

• Solving: leads to or .

• Example: Solve .

Graphs and Functions

Graphing Linear Equations (Point Plotting)

Graphing linear equations helps visualize solutions and relationships between variables.

• Linear Equation:

• Point Plotting: Choose values for , compute , and plot points.

• Example: Plot for .

Intro to Slope & Intro to Functions

Slope measures the steepness of a line, and functions describe relationships between inputs and outputs.

• Slope Formula:

• Function: A rule assigning each input exactly one output, .

• Example: Find the slope between and .

More Graphing & Writing Equations of Lines

Writing equations of lines is essential for modeling linear relationships.

• Slope-Intercept Form:

• Point-Slope Form:

• Example: Write the equation of a line with slope $3(2,4)$.

Scatter Plots

Scatter plots display data points and help identify relationships between variables.

• Definition: A graph of individual data points.

• Application: Used to analyze trends and correlations.

Systems of Linear Equations

Solving Systems by Graphing & Substitution

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

• Graphing: Plot both equations; intersection is the solution.

• Substitution: Solve one equation for a variable, substitute into the other.

• Example: Solve and .

Solving Systems with Elimination, Types of Solutions, Applications

The elimination method combines equations to eliminate a variable. Systems can have one, none, or infinitely many solutions.

• Elimination: Add or subtract equations to eliminate a variable.

• Types of Solutions: Consistent (one), inconsistent (none), dependent (infinite).

• Example: Solve and .

Exponents, Polynomials, and Polynomial Functions

Laws of Exponents

Exponent rules simplify expressions involving powers.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Example: Simplify .

Polynomials

Polynomials are expressions with terms consisting of variables raised to whole-number exponents.

• Definition:

• Degree: Highest exponent of the variable.

• Example:

Dividing Polynomials

Polynomials can be divided using long division or synthetic division.

• Long Division: Divide each term sequentially.

• Example: Divide by .

Factoring

GCF & Factor by Grouping

Factoring is the process of writing a polynomial as a product of simpler polynomials.

• GCF: Greatest common factor of all terms.

• Factor by Grouping: Group terms to factor common elements.

• Example: Factor .

Factoring AC Method

The AC method is used for factoring quadratics of the form .

• Steps: Multiply , find factors that sum to , rewrite and factor by grouping.

• Example: Factor .

Quadratic Equations and Functions

Solving Quadratics by Factoring & Square Root Property

Quadratic equations can be solved by factoring or using the square root property.

• Standard Form:

• Square Root Property: If , then

• Example: Solve .

Completing the Square & Quadratic Formula

Completing the square and the quadratic formula are methods for solving any quadratic equation.

• Completing the Square: Transform into .

• Quadratic Formula:

• Example: Solve using the quadratic formula.

Rational Expressions and Functions

Adding, Subtracting, Multiplying, Dividing Rational Functions

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

• Definition: where

• Operations: Find common denominators for addition/subtraction; multiply/divide as usual.

• Example: Add .

Practice Exams and Quizzes

Practice midterms and finals, as well as quizzes, are provided to reinforce understanding and prepare for assessments.

• Purpose: Review and apply concepts from all course topics.

• Strategy: Practice solving problems from each topic area.

Summary Table: Main Course Topics

Additional info: The assignment list closely matches the standard Intermediate Algebra curriculum, covering all major topics required for college-level algebra proficiency.