College Algebra Practice Exam Study Notes

Study Guide - Smart Notes

Tailored notes based on your materials, expanded with key definitions, examples, and context.

Equations & Inequalities

Simplifying Rational Expressions

Rational expressions are fractions in which the numerator and/or denominator are polynomials. Simplifying involves factoring and reducing common factors.

Key Point: Factor both numerator and denominator completely.
Key Point: Cancel any common factors.
Example: Simplify Factor:

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable has an exponent of one.

Key Point: Isolate the variable using inverse operations.
Key Point: Check your solution by substituting back into the original equation.
Example: Solve Subtract 5: Divide by 2:

Solving and Graphing Inequalities

Inequalities compare two expressions using symbols such as <, >, ≤, or ≥. Solutions are often represented on a number line or using interval notation.

Key Point: When multiplying or dividing both sides by a negative number, reverse the inequality sign.
Key Point: Express solutions in interval notation.
Example: Solve Add 4: Interval notation:

Review of Algebra

Order of Operations

Order of operations ensures that mathematical expressions are evaluated correctly. The standard order is Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS).

Key Point: Always perform operations inside parentheses first.
Key Point: Exponents come before multiplication and division.
Example: Simplify First, Then, Next, Finally,

Properties of Real Numbers

Real numbers have several important properties, including closure, commutativity, associativity, distributivity, identity, and inverses.

Key Point: The sum or product of any two real numbers is a real number (closure).
Key Point: (commutative property of addition).
Key Point: (distributive property).

Functions

Function Notation and Evaluation

A function is a relation that assigns exactly one output to each input. Function notation uses to denote the output when is the input.

Key Point: To evaluate a function, substitute the input value for the variable.
Example: If , then

Identifying Functions from Equations

Equations can be rewritten in function notation to clarify the relationship between variables.

Key Point: If each input corresponds to exactly one output, the relation is a function.
Example: Rewrite as

Graphs of Equations

Matching Equations to Graphs

Graphs visually represent equations. Recognizing the shape and key features helps match equations to their graphs.

Key Point: Quadratic equations produce parabolas.
Key Point: The vertex and direction (up/down) are determined by the coefficients.
Example: is a parabola opening upward with vertex at

Polynomial Functions

Evaluating Polynomial Functions

Polynomial functions are expressions involving sums of powers of with coefficients. To evaluate, substitute the given value for .

Key Point: Use substitution and order of operations.
Example: Find for

Systems of Equations

Solving Systems by Substitution

Systems of equations involve finding values that satisfy all equations simultaneously. Substitution is one method for solving.

Key Point: Solve one equation for one variable, then substitute into the other equation.
Example: Solve and Add equations: Substitute:

Applications

Word Problems and Modeling

Algebra is used to model real-world situations, such as cost, distance, and time problems. Translate the problem into equations and solve.

Key Point: Define variables clearly and set up equations based on the scenario.
Example: If Sally can paint a room in 7 hours and Elaine in 4 hours, their combined rate is rooms per hour.

Radicals and Exponents

Simplifying Radical Expressions

Radical expressions involve roots, such as square roots. Simplifying may involve combining like terms or rationalizing denominators.

Key Point:
Key Point: for
Example: Simplify

Properties of Exponents

Exponents indicate repeated multiplication. Key properties help simplify expressions.

Key Point:
Key Point:
Key Point: for

Tables

Comparison of Solution Methods

Some questions involve choosing the correct method or solution from a list. Below is a sample table comparing solution methods for equations:

Method	When to Use	Example
Substitution	One equation is easily solved for one variable	,
Elimination	Coefficients of one variable are opposites	,
Graphing	Visual solution needed	Plot and

Additional info: Table inferred for academic completeness.