Rational Equations

Solving Rational Equations

Rational equations involve fractions with polynomials in the numerator and denominator. Solving these requires careful handling of denominators and checking for extraneous solutions.

• Step 1: Find the Least Common Denominator (LCD) of all terms.

• Step 2: Multiply each term by the LCD and simplify.

• Step 3: Solve for x.

• Step 4: Check solutions against exclusions (values that make any denominator zero).

Exclusions: Set each denominator not equal to zero and solve for excluded values.

Example: Solve

Find LCD, multiply through, solve, and check for excluded values.

Quadratic Equations

Quadratic Formula

The quadratic formula solves equations of the form .

• Standard Form:

• Formula:

Example: Solve using the formula.

Factoring Quadratics

Factoring is another method to solve quadratic equations.

• Step 1: Find the Greatest Common Factor (GCF).

• Step 2: Factor the quadratic (may require multiple steps).

• Step 3: Set each factor equal to zero and solve for x.

• Step 4: Check solutions.

Radical Equations

Solving Radical Equations

Radical equations contain variables under a root. Isolating the radical and squaring both sides is key.

• Step 1: Isolate the radical.

• Step 2: Square both sides.

• Step 3: Solve for x.

• Step 4: Check for extraneous solutions.

Absolute Value Equations and Inequalities

Solving Absolute Value Equations

Absolute value equations require splitting into two cases.

• Step 1: Isolate the absolute value expression.

• Step 2: Write two equations: one with positive, one with negative right side.

• Step 3: Solve for x.

• Step 4: Interpret solutions based on the value of c (positive: two solutions, zero: one solution, negative: no solution).

Solving Absolute Value Inequalities

Absolute value inequalities are handled differently based on the inequality sign.

• For ≥ or >: Write two inequalities using ±c.

• For ≤ or <: Write a compound inequality using ±c.

Solving Inequalities

Linear Inequalities

Solving linear inequalities involves similar steps to equations, but solutions are often written in interval notation.

• Step 1: Combine like terms and isolate x on the left, constants on the right.

• Step 2: Solve for x.

• Step 3: Express the solution in interval notation.

Functions

Definition and Types

A function is a relation where each input x corresponds to exactly one output y.

• One-to-One Function: Each y value is paired with only one x.

• Piecewise Function: Defined by different expressions depending on the value of x.

Example:

Finding Domain

The domain is the set of all possible input values for a function.

• Step 1: Assume all real numbers ().

• Step 2: Identify restrictions:

  • x in denominator: Set denominator ≠ 0 and solve.

  • x under even root: Set radicand ≥ 0 and solve.

• Step 3: Write domain in interval notation.

Algebra of Functions

Operations and Compositions

Functions can be added, subtracted, multiplied, divided, and composed.

• Step 1: Write operation as definition.

• Step 2: Substitute values.

• Step 3: Distribute and combine like terms.

• Step 4: Solve as needed.

Composite Functions

A composite function is formed by applying one function to the result of another: .

• Step 1: Write as definition.

• Step 2: Substitute values.

• Step 3: Simplify.

Decomposing Composite Functions

Decomposition involves identifying the inner and outer functions.

• Step 1: Identify (the more complex part).

• Step 2: Identify (replace with ).

Inverse Functions

The inverse function reverses the effect of .

• Step 1: Replace with .

• Step 2: Switch and .

• Step 3: Solve for .

• Step 4: Replace with .

Quadratic Functions and Parabolas

Vertex and Axis of Symmetry

The vertex of a parabola is its maximum or minimum point. The axis of symmetry passes through the vertex.

• Vertex: where and

• Axis of Symmetry:

Example: For , ,

Graphing Parabolas

• a > 0: Parabola opens upward (smiley face).

• a < 0: Parabola opens downward (frowny face).

Systems of Equations and Determinants

Solving Systems of Linear Equations

Systems can be solved by substitution, addition (elimination), or matrix methods.

• Substitution: Use when a variable has no coefficient.

• Addition: Use when you can create opposites.

• Step 1: Solve for one variable.

• Step 2: Substitute into the other equation.

• Step 3: Solve for the second variable.

• Step 4: Write as an ordered pair and check.

Determinants and Cramer's Rule

Determinants are used to solve systems using matrices.

• First and Second Order Determinant:

• Cramer's Rule:

Third-Order Determinant:

Exponential and Logarithmic Functions

Logarithmic and Exponential Forms

Logarithms and exponentials are inverse operations.

• Logarithmic Form:

• Exponential Form:

Change of Base Formula:

Properties of Logarithms

• Multiplication:

• Division:

• Powers:

Order for Expansion: 1. Quotients/Products 2. Powers

Order for Condensing: 1. Powers 2. Products/Quotients

Solving Logarithmic Equations

• Step 1: Condense to one log = one log.

• Step 2: Set arguments equal:

• Step 3: Solve for x.

• Step 4: Check that arguments are positive.

Solving Exponential Equations

• Step 1: Isolate exponential expression.

• Step 2: Take log or ln of both sides.

• Step 3: Use power rule to simplify.

• Step 4: Solve for x.

• Step 5: Check solution.

Example:

Compound Interest

Formulas

Compound interest is calculated using exponential functions.

• General Formula:

• Continuous Compounding:

Example: , , ,

Systems of Linear Inequalities

Graphing Linear Inequalities

Graphing linear inequalities involves shading regions that satisfy the inequality.

• Step 1: Replace inequality with equality and graph the line.

• Step 2: Use solid line for ≤ or ≥, dashed for < or >.

• Step 3: Choose a test point not on the line.

• Step 4: Substitute test point; if true, shade that region.

Solving by Addition (Elimination)

• Step 1: Create opposites for variables.

• Step 2: Add equations to eliminate one variable.

• Step 3: Solve for remaining variable.

• Step 4: Substitute back and solve for other variable.

Graphing Multiple Inequalities

Graph each inequality separately, then shade the region that satisfies all.

Matrix Methods

Gauss-Jordan Elimination

Gauss-Jordan elimination is a systematic method for solving systems using matrices.

• Step 1: Build the augmented matrix.

• Step 2: Use row operations to reduce to row-echelon form.

• Step 3: Solve for variables.

Cramer's Rule (2x2 and 3x3)

See determinants section above for formulas.

• Step 1: Build matrix.

• Step 2: Find determinant .

• Step 3: Find and .

• Step 4: Compute , .

• Step 5: Write as ordered pair and check solution.

Third-Order Determinant

For a 3x3 matrix:

Additional info: These notes cover all major topics in college algebra, including equations, inequalities, functions, systems, matrices, determinants, and exponential/logarithmic functions. For exam preparation, practice solving each type of equation and graphing functions and inequalities.