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Algebra I / Elem. Algebra

The Basics

1.1 Intro to Algebra
1.2 Real Number System and Calculators

Linear Equations and Inequalities

Functions and Graphs I

3.1^s Introduction to Graphing
3.2 Solving Equations Graphically
3.3 Functions I - Introduction
3.4 Functions II - Domain, Range, & Operations

Lines and thier Graphs

4.1 Intercept Points
4.2^s Slopes
4.3 Equations of Lines
4.4 Graphing Linear Inequalities

Algebra II / E&I Algebra

Linear Systems

5.1 2x2 Systems of Equations I - Graphically
5.2 2x2 Systems of Equations II - Substitution Method
5.3 2x2 Systems of Equations III - Elimination Method
5.4^s 3x3 Systems of Equations
5.5 Applications
5.6 Systems of Linear Inequalities

Exponents & Polynomials

Intermediate Algebra starts here!

Factoring

Rational Expressions

Rational Equations and Applications

Algebra III / Inter. Algebra

Radical Expressions

10.1 Roots & Radicals
10.2 Fractional Exponents
10.3 Operations I - Simplify & Multiply
10.4 Operations II - Add, Subtract, & Divide
10.5 Operations III - Distribute, Foil, & Factor
10.6^s Complex Numbers

Nonlinear Equations and Applications

11.1 Radical Equations
11.2 Quadratic Equations I - Root Method
11.3^s Quadratic Equations II - Complete the Square
11.4^s Quadratic Equations III - Quadratic Formula
11.5^s Applications - Geometry & Pythagorean Theorem

Functions and Graphs II

12.1 Parabolas I - Algebraic Approach
12.2 Pareabolas II - Graphical Approach
12.3^s Functions III - Composite & Inverse

Exponential and Logarithmic Functions

13.1^s Exponential Functions & Graphs
13.2 Logarithmic Functions & Graphs
13.3 Exponential & Logarithmic Equations

Liberal Arts Maths

Under Construction

College Algebra

Equations and Inequalitites

Functions and Graphs

Polynomial and Rational Functions

3.1 Polynomial Functions I - Introduction, Zeros
3.2 Polynomial Functions II - Division, Remainder Thm, Factor Thm
3.3 Polynomial Functions III - Complex Zeros, Fundamental Thm of Algebra, Descartes Rule
3.4 Rational Functions & Graphs
3.5 Partial Fractions

Exponential and Logarithmic Functions

Systems and Matrices

Conic Sections

Sequences and Series

7.1^s Introduction to Sequences and Series
7.2 Arithmetic Sequences and Series
7.3 Geometric Sequences and Series

Trigonometry

The Six Trigonometric Functions

Right Triangle Trigonometry

Circular Functions

Graphs of Trigonometric Functions

4.1^s Basic Graphs
4.2^s Amplitude & Period
4.3^s Vertical Translations & Phase Shift
4.4 General Equation & Graph to Equation
4.5^s Combination Graphs & Harmonic Motion
4.6^s Inverse Trigonometric Functions

Trigonometric Identities

5.1 Proving Trig Identities
5.2 Sum/Difference Identities
5.3 Double-Angle Identities
5.4^s Half-Angle Identities
5.5 IDs Involving Inverse Trig Functions

Trigonometric Equations

Oblique Triangles and the Laws

Vectors

Complex, Parametric, and Polar Forms

9.1 Trigonometric Form of Complex Numbers
9.2 Products and Quotients in Trig Form
9.3 Powers and Roots in Trig Form
9.4 Parametric Equations and Graphs
9.5^s Polar Equations and Graphs

Calculus I

Limits and Continuity

Derivatives

2.1 Definition of Derivative
2.2 Derivatives of Basic Functions
2.3 Motion and the Derivative
2.4 Product Rule, Quotient Rules, & Deriv. of Trig Functions
2.5 Chain Rule
2.6 Implicit Differentiation
2.7 Derivatives of Exponential and Logarithmic Functions
2.8 Logarithmic Differentiation
2.9^s Derivatives of Inverse Functions
2.10^s Derivatives of Inverse Trig Functions
2.11 Related Rate Applications

Analysis of Curves

3.1 Extreme Values
3.2 Mean Value Theorem for Derivatives
3.3 First Derivative Test & Increasing-Decreasing Functions
3.4 Second Derivative Test & Concavity
3.5^s Curve Sketching
3.6 Fun with Reform Calculus
3.7 Optimization Applications

Antiderivatives

Calculus II

Transcendental Functions

5.0^s Review of Transcendental Functions and thier Derivatives
5.1 Integrals Resulting in Logarithmic Functions
5.2 Integrals Resulting in Inverse Trig Functions
5.3 Hyperbolic Trigonometric Functions
5.4 Numeric and Electronic Integration

Geometry

Physics

Integration Techniques

Calculus of Infinity

Parametric, Polar, and Conic Curves

Calculus III

Under Construction

S = contains supplemental resources

Course: Algebra II / Intermediate Algebra
Topic: Rational Equations and Applications
Subtopic: Rational Equations

Overview

Solving rational equations is the focus of this lesson. It is important to recognize the difference between a rational equation and a rational expression. With equations we can actually solve for x and since there is an equals sign we are able to multiply both sides by something to eliminate the denominators. This wasn't true with expressions. When working with rational expressions all we could do, for example, is add them together by building up the denominators to be the same -- you could not multiply through by the LCD and eliminate it.

We will solve two types of rational equations. Proportions (single fraction = single fraction) are solved by the cross products method. More complicated rational expressions, ones containing more than two terms, are solved by the LCD method where the entire expression is multiplied through by the LCD to clear all the denominators.

Objectives

By the end of this topic you should know and be prepared to be tested on:

9.1.1 Know the difference between a rational expression and a rational equation
9.1.2 Recognize a proportion
9.1.3 Solve rational equations that are proportions by the cross products method
9.1.4 Find the LCD of a rational equation
9.1.5 Solve rational equations that are not proportions by the LCD method
9.1.6 Find domain restrictions of rational equations
9.1.7 Recognize when an apparent solution to a rational equation is actually a domain restriction to the original equation

Terminology

Define: rational equation, proportion, cross products method (a.k.a. means and extremes)

Text Notes
Text: Intro & Inter Algebra for CS 3ed by Blitzer, sect. 7.6

When solving a rational equation, concentrate mostly on the algebraic approach. Solving graphically is covered more in a pre-calculus class. You may opt to use the graph to check your answers if you would like (see my Calculator Guide: Intersection Points for detailed steps).
Pg 511 has an important "study tip" box that compares simplifying rational expressions to solving rational equations. I can't stress how important it is to differentiate expressions from equations and to know what to do with each!
When finding the domain restriction, concentrate on the algebraic approach (finding the x-values that make the denominators in the original equation zero) not the graphical approach (which represents domain restrictions of rational expressions as vertical asymptote lines). Always make sure that you do not include any domain restrictions in your final list of solutions. This text calls this step "checking the proposed solutions".
Material starting on the bottom of page 512 with "Solving a Formula for a Variable" is covered in a later lecture notes.

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