Ch. 1: Equations, Inequalities, and Applications

Linear Equations

Linear equations are fundamental in algebra and involve variables raised only to the first power. The goal is to isolate the variable to find its value.

• Distributive Property: Apply to remove parentheses.

• Combine Like Terms: Group similar variable and constant terms.

• Isolate Variable: Move variables and constants to opposite sides as needed.

• Check Solution: Substitute back into the original equation to verify.

Example: Solve Distribute: Subtract 6: Divide by 2:

Equations with Fractions

To simplify equations with fractions, multiply both sides by the Least Common Denominator (LCD).

• LCD: The smallest number that all denominators divide into.

• Clear Fractions: Multiply every term by the LCD before solving.

Example: LCD is 6: Subtract 2: Divide by 3:

Types of Solutions

• Conditional: One solution.

• Identity: Infinitely many solutions (true for all values).

• Inconsistent: No solution (contradiction).

Rational Equations

Equations involving fractions with variables in the denominator.

• Restrictions: Denominator cannot be zero.

• Multiply by LCD: Clear denominators.

• Extraneous Solutions: Check for and reject any that make the denominator zero.

Complex Numbers

Complex numbers extend real numbers to include the imaginary unit .

• Definition:

• General Form:

• Powers of : Cycle every 4: , , ,

• Division: Use conjugates to rationalize denominators.

Quadratic Equations

Quadratic equations have the form .

• Factoring: Express as a product of binomials.

• Square Root Property: If , then .

• Completing the Square: Rearrange to .

• Quadratic Formula:

• Discriminant: determines the number and type of solutions.

Example: Solve Factoring: Solutions: ,

Inequalities

Inequalities are solved similarly to equations, but the direction of the inequality must be reversed if multiplying or dividing by a negative number.

• Interval Notation: Used to express solution sets.

• Absolute Value: Split into two cases for equations and compound inequalities for inequalities.

Example: Solve Compound inequality: Add 2:

Ch. 2: The Rectangular Coordinate System, Lines, and Circles

Ordered Pairs and Quadrants

The coordinate plane consists of ordered pairs , divided into four quadrants.

• x-intercept: Where

• y-intercept: Where

Slope of a Line

Slope measures the steepness of a line.

• Formula:

• Horizontal Lines:

• Vertical Lines: Slope is undefined

Forms of Linear Equations

• Slope-Intercept Form:

• Point-Slope Form:

• Standard Form:

• Parallel Lines: Same slope

• Perpendicular Lines: Slopes are negative reciprocals

Circles

The equation of a circle in the coordinate plane is:

• Standard Form:

• Completing the Square: Used to convert general form to standard form

Ch. 3: Functions

Definition and Notation

A function assigns exactly one output to each input. The vertical line test determines if a graph represents a function.

• Function Notation:

Domain and Range

• Domain: All possible input values (x-values)

• Range: All possible output values (y-values)

• Restrictions: Values that make denominators zero or expressions under radicals negative are excluded

Function Operations

• Add/Subtract/Multiply: ,

• Quotient: , domain excludes

Transformations

• Shifts: moves the graph horizontally and vertically

• Reflections: Over axes

• Stretches/Compressions: Change the shape of the graph

Composition of Functions

• Definition:

• Domain: Must consider both and

Ch. 4: Polynomial and Rational Functions

Polynomials

Polynomials are smooth, continuous functions. Their end behavior depends on the leading term.

• Leading Term:

• Degree: Even degree: ends go same direction; odd degree: ends go opposite directions

• Zeros: Found by factoring; x-intercepts

• Multiplicity: Even: graph bounces; Odd: graph crosses

• Turning Points: Maximum is for degree

Rational Functions

• Domain: Excludes values where denominator is zero

• Simplify: Only after noting restrictions

• Vertical Asymptotes: Denominator equals zero after canceling common factors

• Holes: Where factors cancel

• Horizontal Asymptotes: Compare degrees of numerator and denominator

Ch. 5: Exponential and Logarithmic Functions and Equations

Exponential Functions

Exponential functions have the form .

• Domain: All real numbers

• Horizontal Asymptote: Usually , shifts may change this

• Growth:

• Decay:

• Constant :

Logarithmic Functions

Logarithms are the inverse of exponentials.

• Conversion:

• Common Log: is base 10

• Natural Log: is base

• Properties:

  • Product:

  • Quotient:

  • Power:

• Solving Equations: Use rewriting or take / of both sides

• Domain: Log arguments must be