The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding or subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate converting rational expressions to equivalent forms with different specified denominators.
55 addition and subtraction of rational expressions alg1testreview
The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding and subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate adding and subtracting rational expressions, as well as converting them to have a common denominator.
The document discusses rules and procedures for adding and subtracting rational expressions. It states that fractions can only be directly added or subtracted if they have the same denominator. It provides an example of adding and subtracting fractions with the same denominator and simplifying the results. It also discusses how to convert fractions with different denominators to have a common denominator before adding or subtracting them, using the least common multiple (LCM) of the denominators. It provides an example problem that demonstrates converting fractions to equivalent forms with a specified common denominator.
48 factoring out the gcf and the grouping methodalg1testreview
The document discusses factoring quantities and finding common factors and greatest common factors (GCF). It provides examples of factoring numbers like 12 and finding the common factors and GCF of expressions. Factoring means rewriting a quantity as a product without 1. A common factor is one that belongs to all quantities, and the GCF is the largest common factor considering coefficients and degrees of factors. Examples show finding the common factors of expressions and the GCF of quantities like 24 and 36.
This document discusses antiderivatives and indefinite integration. It explains that an antiderivative is a function whose derivative is equal to a given function, and that the general solution to a differential equation involving antiderivatives contains an arbitrary constant of integration. It provides examples of finding antiderivatives using basic integration rules and rewriting integrands before integrating. The key points are that antiderivatives are defined up to an additive constant, and that rewriting the integrand is an important step in the integration process.
This document discusses antiderivatives and indefinite integration. It explains that an antiderivative is a function whose derivative is equal to a given function, and that the general solution to a differential equation involving antiderivatives contains an arbitrary constant of integration. It provides examples of finding antiderivatives using basic integration rules and rewriting integrands before integrating. The key points are that antiderivatives are defined up to an additive constant, and that rewriting the integrand is an important step in the integration process.
This document discusses antiderivatives and indefinite integration. It explains that an antiderivative is a function whose derivative is equal to a given function, and that the general solution to a differential equation involving antiderivatives contains an arbitrary constant of integration. It provides examples of finding antiderivatives using basic integration rules and rewriting integrands before integrating. The key points are that antiderivatives are defined up to an additive constant, and that rewriting the integrand is an important step in the integration process.
16 the multiplier method for simplifying complex fractionselem-alg-sample
The document discusses two methods for simplifying complex fractions. A complex fraction is a fraction with fractions in the numerator or denominator. The first method reduces the complex fraction to an "easy" regular division problem by combining fractions in the numerator and denominator. The second method multiplies the lowest common denominator of all terms to the numerator and denominator to simplify. An example using each method is provided.
MIT Math Syllabus 10-3 Lesson 3: Rational expressionsLawrence De Vera
This document provides information about rational expressions including:
- A rational expression is a fraction where the numerator and denominator are polynomials.
- The domain of a rational expression excludes any values that would cause division by zero.
- Rational expressions can be simplified by factoring the numerator and denominator and cancelling out common factors.
- The four arithmetic operations can be performed on rational expressions using similar properties as rational numbers.
- Complex fractions containing fractions in the numerator or denominator can be simplified by finding the least common denominator or multiplying the numerator and denominator by the reciprocal.
- Rational expressions can be used to solve applications involving averages, rates, or other fractional relationships.
55 addition and subtraction of rational expressions alg1testreview
The document discusses addition and subtraction of rational expressions. It states that rational expressions can only be added or subtracted if they have the same denominator. It provides the rule for adding and subtracting rational expressions with the same denominator. It also discusses converting rational expressions to have a common denominator so they can be added or subtracted, using the least common multiple of the denominators. Examples are provided to demonstrate adding and subtracting rational expressions, as well as converting them to have a common denominator.
The document discusses rules and procedures for adding and subtracting rational expressions. It states that fractions can only be directly added or subtracted if they have the same denominator. It provides an example of adding and subtracting fractions with the same denominator and simplifying the results. It also discusses how to convert fractions with different denominators to have a common denominator before adding or subtracting them, using the least common multiple (LCM) of the denominators. It provides an example problem that demonstrates converting fractions to equivalent forms with a specified common denominator.
48 factoring out the gcf and the grouping methodalg1testreview
The document discusses factoring quantities and finding common factors and greatest common factors (GCF). It provides examples of factoring numbers like 12 and finding the common factors and GCF of expressions. Factoring means rewriting a quantity as a product without 1. A common factor is one that belongs to all quantities, and the GCF is the largest common factor considering coefficients and degrees of factors. Examples show finding the common factors of expressions and the GCF of quantities like 24 and 36.
This document discusses antiderivatives and indefinite integration. It explains that an antiderivative is a function whose derivative is equal to a given function, and that the general solution to a differential equation involving antiderivatives contains an arbitrary constant of integration. It provides examples of finding antiderivatives using basic integration rules and rewriting integrands before integrating. The key points are that antiderivatives are defined up to an additive constant, and that rewriting the integrand is an important step in the integration process.
This document discusses antiderivatives and indefinite integration. It explains that an antiderivative is a function whose derivative is equal to a given function, and that the general solution to a differential equation involving antiderivatives contains an arbitrary constant of integration. It provides examples of finding antiderivatives using basic integration rules and rewriting integrands before integrating. The key points are that antiderivatives are defined up to an additive constant, and that rewriting the integrand is an important step in the integration process.
This document discusses antiderivatives and indefinite integration. It explains that an antiderivative is a function whose derivative is equal to a given function, and that the general solution to a differential equation involving antiderivatives contains an arbitrary constant of integration. It provides examples of finding antiderivatives using basic integration rules and rewriting integrands before integrating. The key points are that antiderivatives are defined up to an additive constant, and that rewriting the integrand is an important step in the integration process.
16 the multiplier method for simplifying complex fractionselem-alg-sample
The document discusses two methods for simplifying complex fractions. A complex fraction is a fraction with fractions in the numerator or denominator. The first method reduces the complex fraction to an "easy" regular division problem by combining fractions in the numerator and denominator. The second method multiplies the lowest common denominator of all terms to the numerator and denominator to simplify. An example using each method is provided.
MIT Math Syllabus 10-3 Lesson 3: Rational expressionsLawrence De Vera
This document provides information about rational expressions including:
- A rational expression is a fraction where the numerator and denominator are polynomials.
- The domain of a rational expression excludes any values that would cause division by zero.
- Rational expressions can be simplified by factoring the numerator and denominator and cancelling out common factors.
- The four arithmetic operations can be performed on rational expressions using similar properties as rational numbers.
- Complex fractions containing fractions in the numerator or denominator can be simplified by finding the least common denominator or multiplying the numerator and denominator by the reciprocal.
- Rational expressions can be used to solve applications involving averages, rates, or other fractional relationships.
This document discusses design matrices and contrast statements in SAS regression procedures. It defines a design matrix as a matrix of explanatory variable values used in regression analyses. It describes three common coding schemes for design matrices in SAS - GLM, effect, and reference (REF) coding - and how they parameterize categorical variables. It provides examples of how these different coding schemes impact odds ratio interpretation and the formulation of contrast statements in logistic regression analyses.
This document provides an overview of rational expressions and equations. It discusses reviewing rational expressions, domains of rational expressions, writing rational expressions in lowest terms, multiplying and dividing rational expressions, complex fractions, adding and subtracting rational expressions, and simplifying complex fractions. Examples are provided to illustrate key concepts and steps for working with rational expressions.
Kahler Differential and Application to Ramification - Ryan Lok-Wing PangRyan Lok-Wing Pang
This document discusses the application of Kähler differentials to the study of ramification in algebraic number theory. It defines Kähler differentials and constructs the module of relative differentials. Properties like exact sequences are proven. The concept of the different ideal is introduced, which encodes ramification data in field extensions. It is shown that the different ideal is the annihilator of the module of Kähler differentials, providing a geometric characterization of ramification.
The document discusses methods for simplifying complex fractions. A complex fraction is a fraction with fractions in the numerator or denominator. The first method is to combine the numerator and denominator into single fractions using cross multiplication. The second method is to multiply the lowest common denominator of all terms to both the numerator and denominator. Examples are provided to demonstrate both methods.
This document discusses dependency preserving decomposition in relational databases. It defines dependency preservation as decomposing a relation such that the set of functional dependencies is preserved. An algorithm is presented to check if a decomposition preserves dependencies by iterating through each dependency and checking if the right hand side is contained within the closure of the left hand side within the decomposed relations. An example is provided to demonstrate how to apply the algorithm to verify a decomposition preserves dependencies.
This document summarizes a lecture on database design theory that covered topics like database design problems, functional dependencies, decomposition, and normalization. It began with an overview of the concepts of redundancy, anomalies, and functional dependencies. It then discussed decomposition rules, lossless joins, dependency preservation, and normal forms. The lecture aimed to explain how to model databases and design relational schemas to minimize redundancy and avoid anomalies.
This document provides an overview of the key concepts covered in Chapter 6 of a mathematics textbook, which includes:
1) Defining rational expressions as ratios of two polynomials and focusing on adding, subtracting, multiplying, and dividing rational expressions.
2) Concluding with solving rational equations and applications of rational equations.
3) Summarizing the chapter sections, which cover topics like rational expressions and functions, operations on rational expressions, and applications of rational equations.
The document discusses rational expressions and operations involving them. It covers evaluating, simplifying, multiplying, dividing, adding and subtracting rational expressions. It also discusses finding least common denominators to add or subtract rational expressions with unlike denominators. The sections include evaluating rational expressions, simplifying rational expressions, multiplying and dividing rational expressions, and adding and subtracting rational expressions with the same or different denominators.
13 multiplication and division of rational expressionselem-alg-sample
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions as the product of the numerators over the product of the denominators. It provides an example of simplifying a rational expression by factoring the top and bottom and canceling like terms. It then gives another example with two parts, simplifying and expanding the answers of rational expression operations.
Deductivereasoning and bicond and algebraic proofsjbianco9910
1. The document discusses biconditional statements, conditional statements, and using deductive reasoning in geometry. It provides examples of identifying conditionals within biconditionals, writing definitions as biconditionals, and solving equations with justification in both algebra and geometry.
2. Key concepts covered include using properties of equality to write algebraic proofs, properties of congruence corresponding to properties of equality, and identifying properties of equality and congruence that justify statements.
3. Examples are provided of solving equations algebraically and geometrically with justification for each step, identifying conditionals within biconditionals, and writing definitions as biconditionals.
MIT Math Syllabus 10-3 Lesson 4: Rational exponents and radicalsLawrence De Vera
This document discusses rational exponents and radicals. It begins by extending the definition of exponents to include rational numbers so that expressions like 21/2 are meaningful. It defines b1/n as the nth root of b. Properties of rational exponents and radicals are discussed, including how to simplify expressions involving rational exponents and radicals. Radicals can be added, subtracted, multiplied, and rationalized using properties similar to exponents.
The document discusses relational database design and normalization. It covers first normal form, functional dependencies, and decomposition. The goal of normalization is to avoid data redundancy and anomalies. First normal form requires attributes to be atomic. Functional dependencies specify relationships between attributes that must be preserved. Decomposition breaks relations into smaller relations while maintaining lossless join properties. Higher normal forms like Boyce-Codd normal form and third normal form further reduce redundancy.
This document provides an overview of Chapter 14 on rational expressions from a developmental mathematics textbook. It covers simplifying, multiplying, dividing, adding, and subtracting rational expressions. It also discusses finding least common denominators and changing rational expressions to equivalent forms with a common denominator in order to add or subtract expressions. The chapter is divided into sections covering specific topics like simplifying rational expressions, multiplying and dividing, adding and subtracting with the same or different denominators, and solving equations with rational expressions. Examples are provided throughout to illustrate the concepts and procedures.
11.a new computational methodology to find appropriateAlexander Decker
This document proposes a new computational methodology for finding appropriate solutions to fuzzy equations of the form A=B, where A and B are known continuous triangular fuzzy numbers and X is an unknown fuzzy number. It introduces definitions of "positive fuzzy number," "negative fuzzy number," and "half-positive and half-negative fuzzy number." Several propositions and theorems are presented along with proofs to show that the solution to such a fuzzy equation can be a positive, negative, or half-positive/half-negative fuzzy number, depending on the values of A and B. Examples are provided to illustrate solving equations where A and B are positive, negative, or half-positive/half-negative fuzzy numbers. The methodology aims to overcome limitations
A new computational methodology to find appropriateAlexander Decker
This document proposes a new computational methodology for finding appropriate solutions to fuzzy equations of the form A=B, where A and B are known fuzzy numbers and X is an unknown fuzzy number. It introduces definitions of positive, negative, and half-positive/half-negative fuzzy numbers. Propositions and theorems are presented to show that the solution to such an equation can be a positive, negative, or half-positive/half-negative fuzzy number depending on the properties of A and B. Examples are provided to illustrate solving equations where A and B are positive, negative, or half-positive/half-negative fuzzy numbers. The methodology aims to overcome limitations in solving equations where X is an unknown fuzzy number.
This document provides instructions for simplifying compound algebraic fractions that have separate operations in the numerator and denominator. It gives examples of compound fractions with addition, subtraction, and multiplication in both the numerator and denominator, and explains how to simplify each type by distributing the operations.
This document provides instructions for simplifying compound algebraic fractions that have separate operations in the numerator and denominator. It gives examples of compound fractions with addition, subtraction, and multiplication in both the numerator and denominator, and explains how to simplify each type by distributing the operations.
The document discusses addition and subtraction of rational expressions. It states that fractions with the same denominator can be directly added or subtracted, while those with different denominators must first be converted to have a common denominator. The document provides an addition/subtraction rule and examples demonstrating how to perform these operations on rational expressions, including converting fractions to equivalent forms with a specified common denominator.
The document defines rational numbers and provides examples. It discusses equality of rational numbers using the rule that ad = bc. It also covers adding, subtracting, multiplying, and dividing rational numbers. Examples are provided to illustrate operations with rational numbers and comparing rational numbers on the number line. The document contains exercises asking the reader to perform operations with rational numbers and compare rational numbers.
This document provides an overview of linear equations and inequalities. It begins by stating the learning objectives which are to solve linear equations and inequalities, and apply them to word problems. Several examples are then shown of solving linear equations by clearing fractions and combining like terms. The concepts of equivalent equations, solving formulas, and solving linear inequalities are also explained. Interval notation is introduced to describe solutions to inequalities. Finally, a multi-step word problem on break-even analysis is presented and solved to demonstrate applying linear equations to applications.
This document defines key terms related to variables and expressions:
- A variable represents an unknown number using a symbol like x or n.
- A variable expression contains variables and numbers with arithmetic operations but no equal sign, like n - 5.
- To evaluate an expression means to substitute numbers for the variables and simplify the result.
This tutorial teaches how to solve multi-step equations with one variable. It defines key terms like variable, equation, coefficient, constant, and inverse operation. It then walks through an example of solving the equation 2x + 7 = 15 in 3 steps: 1) subtracting 7 from both sides to eliminate the constant, 2) dividing both sides by 2 to isolate the variable x, and 3) determining that x = 4. The tutorial concludes with a quiz to test the learner's understanding of these concepts.
This document discusses design matrices and contrast statements in SAS regression procedures. It defines a design matrix as a matrix of explanatory variable values used in regression analyses. It describes three common coding schemes for design matrices in SAS - GLM, effect, and reference (REF) coding - and how they parameterize categorical variables. It provides examples of how these different coding schemes impact odds ratio interpretation and the formulation of contrast statements in logistic regression analyses.
This document provides an overview of rational expressions and equations. It discusses reviewing rational expressions, domains of rational expressions, writing rational expressions in lowest terms, multiplying and dividing rational expressions, complex fractions, adding and subtracting rational expressions, and simplifying complex fractions. Examples are provided to illustrate key concepts and steps for working with rational expressions.
Kahler Differential and Application to Ramification - Ryan Lok-Wing PangRyan Lok-Wing Pang
This document discusses the application of Kähler differentials to the study of ramification in algebraic number theory. It defines Kähler differentials and constructs the module of relative differentials. Properties like exact sequences are proven. The concept of the different ideal is introduced, which encodes ramification data in field extensions. It is shown that the different ideal is the annihilator of the module of Kähler differentials, providing a geometric characterization of ramification.
The document discusses methods for simplifying complex fractions. A complex fraction is a fraction with fractions in the numerator or denominator. The first method is to combine the numerator and denominator into single fractions using cross multiplication. The second method is to multiply the lowest common denominator of all terms to both the numerator and denominator. Examples are provided to demonstrate both methods.
This document discusses dependency preserving decomposition in relational databases. It defines dependency preservation as decomposing a relation such that the set of functional dependencies is preserved. An algorithm is presented to check if a decomposition preserves dependencies by iterating through each dependency and checking if the right hand side is contained within the closure of the left hand side within the decomposed relations. An example is provided to demonstrate how to apply the algorithm to verify a decomposition preserves dependencies.
This document summarizes a lecture on database design theory that covered topics like database design problems, functional dependencies, decomposition, and normalization. It began with an overview of the concepts of redundancy, anomalies, and functional dependencies. It then discussed decomposition rules, lossless joins, dependency preservation, and normal forms. The lecture aimed to explain how to model databases and design relational schemas to minimize redundancy and avoid anomalies.
This document provides an overview of the key concepts covered in Chapter 6 of a mathematics textbook, which includes:
1) Defining rational expressions as ratios of two polynomials and focusing on adding, subtracting, multiplying, and dividing rational expressions.
2) Concluding with solving rational equations and applications of rational equations.
3) Summarizing the chapter sections, which cover topics like rational expressions and functions, operations on rational expressions, and applications of rational equations.
The document discusses rational expressions and operations involving them. It covers evaluating, simplifying, multiplying, dividing, adding and subtracting rational expressions. It also discusses finding least common denominators to add or subtract rational expressions with unlike denominators. The sections include evaluating rational expressions, simplifying rational expressions, multiplying and dividing rational expressions, and adding and subtracting rational expressions with the same or different denominators.
13 multiplication and division of rational expressionselem-alg-sample
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions as the product of the numerators over the product of the denominators. It provides an example of simplifying a rational expression by factoring the top and bottom and canceling like terms. It then gives another example with two parts, simplifying and expanding the answers of rational expression operations.
Deductivereasoning and bicond and algebraic proofsjbianco9910
1. The document discusses biconditional statements, conditional statements, and using deductive reasoning in geometry. It provides examples of identifying conditionals within biconditionals, writing definitions as biconditionals, and solving equations with justification in both algebra and geometry.
2. Key concepts covered include using properties of equality to write algebraic proofs, properties of congruence corresponding to properties of equality, and identifying properties of equality and congruence that justify statements.
3. Examples are provided of solving equations algebraically and geometrically with justification for each step, identifying conditionals within biconditionals, and writing definitions as biconditionals.
MIT Math Syllabus 10-3 Lesson 4: Rational exponents and radicalsLawrence De Vera
This document discusses rational exponents and radicals. It begins by extending the definition of exponents to include rational numbers so that expressions like 21/2 are meaningful. It defines b1/n as the nth root of b. Properties of rational exponents and radicals are discussed, including how to simplify expressions involving rational exponents and radicals. Radicals can be added, subtracted, multiplied, and rationalized using properties similar to exponents.
The document discusses relational database design and normalization. It covers first normal form, functional dependencies, and decomposition. The goal of normalization is to avoid data redundancy and anomalies. First normal form requires attributes to be atomic. Functional dependencies specify relationships between attributes that must be preserved. Decomposition breaks relations into smaller relations while maintaining lossless join properties. Higher normal forms like Boyce-Codd normal form and third normal form further reduce redundancy.
This document provides an overview of Chapter 14 on rational expressions from a developmental mathematics textbook. It covers simplifying, multiplying, dividing, adding, and subtracting rational expressions. It also discusses finding least common denominators and changing rational expressions to equivalent forms with a common denominator in order to add or subtract expressions. The chapter is divided into sections covering specific topics like simplifying rational expressions, multiplying and dividing, adding and subtracting with the same or different denominators, and solving equations with rational expressions. Examples are provided throughout to illustrate the concepts and procedures.
11.a new computational methodology to find appropriateAlexander Decker
This document proposes a new computational methodology for finding appropriate solutions to fuzzy equations of the form A=B, where A and B are known continuous triangular fuzzy numbers and X is an unknown fuzzy number. It introduces definitions of "positive fuzzy number," "negative fuzzy number," and "half-positive and half-negative fuzzy number." Several propositions and theorems are presented along with proofs to show that the solution to such a fuzzy equation can be a positive, negative, or half-positive/half-negative fuzzy number, depending on the values of A and B. Examples are provided to illustrate solving equations where A and B are positive, negative, or half-positive/half-negative fuzzy numbers. The methodology aims to overcome limitations
A new computational methodology to find appropriateAlexander Decker
This document proposes a new computational methodology for finding appropriate solutions to fuzzy equations of the form A=B, where A and B are known fuzzy numbers and X is an unknown fuzzy number. It introduces definitions of positive, negative, and half-positive/half-negative fuzzy numbers. Propositions and theorems are presented to show that the solution to such an equation can be a positive, negative, or half-positive/half-negative fuzzy number depending on the properties of A and B. Examples are provided to illustrate solving equations where A and B are positive, negative, or half-positive/half-negative fuzzy numbers. The methodology aims to overcome limitations in solving equations where X is an unknown fuzzy number.
This document provides instructions for simplifying compound algebraic fractions that have separate operations in the numerator and denominator. It gives examples of compound fractions with addition, subtraction, and multiplication in both the numerator and denominator, and explains how to simplify each type by distributing the operations.
This document provides instructions for simplifying compound algebraic fractions that have separate operations in the numerator and denominator. It gives examples of compound fractions with addition, subtraction, and multiplication in both the numerator and denominator, and explains how to simplify each type by distributing the operations.
The document discusses addition and subtraction of rational expressions. It states that fractions with the same denominator can be directly added or subtracted, while those with different denominators must first be converted to have a common denominator. The document provides an addition/subtraction rule and examples demonstrating how to perform these operations on rational expressions, including converting fractions to equivalent forms with a specified common denominator.
The document defines rational numbers and provides examples. It discusses equality of rational numbers using the rule that ad = bc. It also covers adding, subtracting, multiplying, and dividing rational numbers. Examples are provided to illustrate operations with rational numbers and comparing rational numbers on the number line. The document contains exercises asking the reader to perform operations with rational numbers and compare rational numbers.
This document provides an overview of linear equations and inequalities. It begins by stating the learning objectives which are to solve linear equations and inequalities, and apply them to word problems. Several examples are then shown of solving linear equations by clearing fractions and combining like terms. The concepts of equivalent equations, solving formulas, and solving linear inequalities are also explained. Interval notation is introduced to describe solutions to inequalities. Finally, a multi-step word problem on break-even analysis is presented and solved to demonstrate applying linear equations to applications.
This document defines key terms related to variables and expressions:
- A variable represents an unknown number using a symbol like x or n.
- A variable expression contains variables and numbers with arithmetic operations but no equal sign, like n - 5.
- To evaluate an expression means to substitute numbers for the variables and simplify the result.
This tutorial teaches how to solve multi-step equations with one variable. It defines key terms like variable, equation, coefficient, constant, and inverse operation. It then walks through an example of solving the equation 2x + 7 = 15 in 3 steps: 1) subtracting 7 from both sides to eliminate the constant, 2) dividing both sides by 2 to isolate the variable x, and 3) determining that x = 4. The tutorial concludes with a quiz to test the learner's understanding of these concepts.
The document provides information on rational numbers and operations involving fractions and integers:
1. It defines rational numbers as numbers that can be represented as fractions a/b where a and b are integers and b is not equal to 0.
2. Rules for addition, subtraction, multiplication, and division of integers and fractions are presented, such as signs of terms determine sign of sum/product.
3. Examples demonstrate applying the rules to evaluate expressions involving integers and fractions.
Online Lecture Chapter R Algebraic Expressionsapayne12
This document provides an overview and examples of skills related to algebraic expressions, including factoring polynomials, simplifying rational expressions, simplifying complex fractions, working with expressions with negative exponents, rationalizing denominators, and rewriting expressions with radicals to have only positive exponents. It includes step-by-step worked examples of simplifying rational expressions and expressions containing radicals. Videos are linked to demonstrate factoring polynomials and rewriting expressions with radicals.
This document provides an overview of rational expressions and operations involving rational expressions. It begins by defining fractional and rational expressions. It then covers performing operations like addition, subtraction, multiplication, and division on rational expressions. This involves using properties of fractions to simplify expressions and cancel common factors. The document also discusses domains, compound fractions, and examples involving calculus concepts.
The document discusses complex numbers, including: defining complex numbers as numbers that can be written in the form a + bi, where a is the real part and b is the imaginary part; operations like addition, subtraction, and multiplication of complex numbers; complex conjugates; dividing complex numbers; and solving quadratic equations that have complex solutions. It provides examples of working through operations with complex numbers and solving a quadratic equation with complex roots.
The document provides lesson material on evaluating variable expressions from algebra. It includes examples of writing variable expressions from word phrases, evaluating expressions by substituting values for variables, and solving multi-step expressions. Students are provided vocabulary for mathematical operations and guided practice problems to work through. There is a reminder for students to bring their math book and calculator to class and that an NWEA test will be taken in the computer lab the next day.
The document discusses various properties of real numbers including the commutative, associative, identity, inverse, zero, and distributive properties. It also covers topics such as combining like terms, translating word phrases to algebraic expressions, and simplifying algebraic expressions. Examples are provided to illustrate each concept along with explanations of key terms like coefficients, variables, and like terms.
This document discusses rational expressions, equations, and inequalities. It defines rational expressions as fractions where the numerator and denominator are polynomials, with the denominator not equal to 0. Examples of rational expressions are provided. Methods for simplifying, adding, subtracting, and multiplying rational expressions are outlined. A rational equation is defined as an equation containing one or more rational expressions. Solving rational equations involves eliminating rational expressions by multiplying both sides by the least common denominator. Steps for solving rational equations are demonstrated through examples. Exercises involving solving rational equations are provided at the end for students to practice.
translating algebraic expression math 7.pptElleMaRie3
The document provides examples for translating word phrases and sentences into mathematical expressions and equations. It includes tables listing common words associated with addition, subtraction, multiplication, division and their corresponding operations. There are also examples of translating phrases and solving equations derived from word sentences. Students are asked to translate phrases and sentences into algebraic expressions and equations.
The document contains information about partial fraction decomposition:
1. It discusses four cases for partial fraction decomposition based on the factors of the denominator: distinct linear factors, repeated linear factors, distinct irreducible quadratic factors, and repeated irreducible quadratic factors.
2. It provides examples to illustrate each case, showing how to set up and solve systems of equations to determine the coefficients of the partial fractions.
3. Homework Task 4 on systems of equations and inequalities is due on August 13, and consultation times for Ms. Durandt are on Thursdays and Fridays at 10:30.
This document introduces the concepts of number theory, including divisibility, greatest common divisors, least common multiples, and modular arithmetic. It defines divisibility as an integer a dividing another integer b if b can be written as a product of a and another integer. The greatest common divisor of two integers is the largest integer that divides both, while the least common multiple is the smallest positive integer divisible by both. Modular arithmetic involves finding the remainder of dividing an integer by a positive integer. Examples are provided to illustrate these key number theory topics.
This document provides an overview of solving linear equations, formulas, and problem solving techniques. It begins by introducing the basic properties of equality used to solve linear equations, such as distributing terms and adding/subtracting terms to isolate the variable. Examples are provided to demonstrate solving equations with fractions and solving literal equations for a specified variable. The document also discusses identities, contradictions, and using a general formula to solve families of linear equations. It concludes by outlining a problem solving guide to organize the steps of reading, visualizing, and developing an equation model to solve word problems.
This document provides examples and explanations for combining like terms in algebraic expressions. It includes examples of combining like terms with one variable, two variables, and using the distributive property to simplify expressions. The examples are followed by a short quiz with 5 problems involving combining like terms and simplifying expressions.
The document discusses ratios, proportions, and solving proportions. It defines a ratio as comparing two numbers by division and a proportion as two equal ratios. Properties of proportions are explored, including the cross products property, exchange property, reciprocal property, and "add one" property. Examples are provided of setting up and solving proportions using these properties.
College Algebra MATH 107 Spring, 2020Page 1 of 11 MA.docxmary772
College Algebra MATH 107 Spring, 2020
Page 1 of 11
MATH 107 FINAL EXAMINATION
This is an open-book exam. You may refer to your text and other course materials as you work
on the exam, and you may use a calculator. You must complete the exam individually.
Neither collaboration nor consultation with others is allowed.
Record your answers and work on the separate answer sheet provided.
There are 30 problems.
Problems #1–12 are Multiple Choice.
Problems #13–21 are Short Answer. (Work not required to be shown)
Problems #22–30 are Short Answer with work required to be shown.
MULTIPLE CHOICE
1. Determine the domain and range of the piecewise function. 1. ______
A. Domain [– 4, 2]; Range [– 2, 4]
B. Domain [– 2, 4]; Range [– 2, 1]
C. Domain [– 2, 4]; Range [– 4 , 2]
D. Domain [0, 2]; Range [– 2, 0]
2. Solve: 3 10x x+ = − 2. ______
A. No solution
B. –2, 5
C. 5
D. –2
-2 2 4 -4
-2
-4
2
4
College Algebra MATH 107 Spring, 2020
Page 2 of 11
3. Determine the interval(s) on which the function is decreasing. 3. ______
A. (–1, 3)
B. (–2, 4)
C. (–3.6, 0) and (6.7, )
D. (–, –2) and (4, )
4. Determine whether the graph of 2y x= + is symmetric with respect to the origin,
the x-axis, or the y-axis. 4. ______
A. not symmetric with respect to the x-axis, not symmetric with respect to the y-axis, and
not symmetric with respect to the origin
B. symmetric with respect to the x-axis only
C. symmetric with respect to the y-axis only
D. symmetric with respect to the origin only
5. Solve, and express the answer in interval notation: | 5 – 6x | 13. 5. ______
A. (–, 3] [−4/3, )
B. (–, −4/3] [3, )
C. [4/3, )
D. [–4/3, 3]
College Algebra MATH 107 Spring, 2020
Page 3 of 11
6. Which of the following represents the graph of 8x − 3y = 24 ? 6. ______
A. B.
C. D.
College Algebra MATH 107 Spring, 2020
Page 4 of 11
7. Write a slope-intercept equation for a line parallel to the line x + 7y = 9 which passes through
the point (28, –3). 7. ______
A. 25y x=− +
B.
1
1
7
y x= − +
C.
1
3
7
y x= − −
D.
1
7
7
y x= −
8. Which of the following best describes the graph? 8. ______
A. It is the graph of a function but not one-to-one.
B. It is the graph of a function and it is one-to-one.
C. It is not the graph of a function.
D. It is the graph of an absolute value relation.
.
College Algebra MATH 107 Spring, 2020
Page 5 of 11
9. Write as an equivalent expression: log (x – 3) – 8 log y + log 1 9. ______
A.
log( 3)
log(8 )
x
y
−
B.
2
log
8
x
y
−
C.
8
3
log
x
y
−
D. ( )log 2 8x y− −
10. Which of the functions corresponds to the graph? 10. ______
A. ( ) 2 xf x e−=
B. ( ) 2 xf x e−= +
C. ( ) 2 xf x e= −
D. ( ) 2xf x e−=
College Algebra MATH 107 Spring, 2020
Page 6 of 11
11. Suppose that for a function f, the equation f (x) = 0 has no real-number solution.
Which of the following statements MUST be tr.
Similar to 15 proportions and the multiplier method for solving rational equations (20)
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- The slope of a line is calculated as the change in the y-values (rise) divided by the change in the x-values (run) between two points on the line.
- This formula is easy to memorize and captures the geometric meaning of slope as the tilt of the line.
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The document discusses methods for finding the least common multiple (LCM) of numbers. It defines a multiple as a number that can be divided evenly by another number. The LCM is the smallest number that is a multiple of all numbers given. Two methods are described: the searching method which tests multiples of the largest number, and the construction method which factors each number and multiplies the highest powers of common factors. Examples are provided to illustrate both methods.
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The document discusses factoring quantities by finding common factors. It defines factoring as rewriting a quantity as a product in a nontrivial way. A quantity is prime if it cannot be written as a product other than 1 times the quantity. To factor completely means writing each factor as a product of prime numbers. Examples show finding common factors of quantities, the greatest common factor (GCF), and extracting common factors from sums and differences using the extraction law.
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2) Division Rule: AN/AK = AN-K
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2. Addition and Subtraction of Rational Expressions
Only fractions with the same denominator may be added or
subtracted directly.
3. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
4. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.
5. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
6. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
7. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
8. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
3
2
9. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
10. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
b. 3x
2x – 3
–
6 – x
2x – 3
11. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
b. 3x
2x – 3
–
6 – x
2x – 3
=
3x – (6 – x)
2x – 3
12. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
b. 3x
2x – 3
–
6 – x
2x – 3
=
3x – (6 – x)
2x – 3
=
3x – 6 + x
2x – 3
13. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
b. 3x
2x – 3
–
6 – x
2x – 3
=
3x – (6 – x)
2x – 3
=
3x – 6 + x
2x – 3
=
4x – 6
2x – 3
14. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
b. 3x
2x – 3
–
6 – x
2x – 3
=
3x – (6 – x)
2x – 3
=
3x – 6 + x
2x – 3
=
4x – 6
2x – 3
=
2(2x – 3)
2x – 3
15. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
b. 3x
2x – 3
–
6 – x
2x – 3
=
3x – (6 – x)
2x – 3
=
3x – 6 + x
2x – 3
=
4x – 6
2x – 3
=
2(2x – 3)
2x – 3
16. Addition and Subtraction of Rational Expressions
Addition and Subtraction Rule
(for rational expressions with the same denominator)
Only fractions with the same denominator may be added or
subtracted directly.
A B
D D
± =
A±B
D
Usually we put the result in the factored form, cancel the
common factor and give the simplified answer.
Example A. Add and subtract and simplify the answer.
a. 5 7
8 8
+ =
5 + 7
8
=
12
8
=
3
2
3
2
b. 3x
2x – 3
–
6 – x
2x – 3
=
3x – (6 – x)
2x – 3
=
3x – 6 + x
2x – 3
=
4x – 6
2x – 3
=
2(2x – 3)
2x – 3
= 2
17. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator.
Addition and Subtraction of Rational Expressions
18. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
19. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier Method
Given the fraction , to convert it into denominator D,
the new numerator is
A
B
A
B
* D.
20. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier Method
Given the fraction , to convert it into denominator D,
the new numerator is
Example B.
a. Convert to a fraction with denominator 12.
A
B
A
B
* D.
5
4
In practice, we write this as
A
B
= A
B
* D D
new numerator
21. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier Method
Given the fraction , to convert it into denominator D,
the new numerator is
Example B.
a. Convert to a fraction with denominator 12.
A
B
A
B
* D.
5
4
In practice, we write this as
A
B
= A
B
* D D
5
4
=
new numerator
22. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier Method
Given the fraction , to convert it into denominator D,
the new numerator is
Example B.
a. Convert to a fraction with denominator 12.
A
B
A
B
* D.
5
4
5
4
* 12
In practice, we write this as
A
B
= A
B
* D D
5
4
= 12
new numerator
new numerator
23. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier Method
Given the fraction , to convert it into denominator D,
the new numerator is
Example B.
a. Convert to a fraction with denominator 12.
A
B
A
B
* D.
5
4
5
4
* 12
3
In practice, we write this as
A
B
= A
B
* D D
5
4
= 12
new numerator
new numerator
24. To add or subtract rational expressions with different
denominators, they have to be converted to expressions with
a common denominator. The easiest common denominator to
work with is their LCM.
Addition and Subtraction of Rational Expressions
Multiplier Method
Given the fraction , to convert it into denominator D,
the new numerator is
Example B.
a. Convert to a fraction with denominator 12.
A
B
A
B
* D.
5
4
5
4
* 12
3 15
12
In practice, we write this as
A
B
= A
B
* D D
5
4
= 12 =
new numerator
25. b. Convert into an expression with denominator 12xy2.
Addition and Subtraction of Rational Expressions
3x
4y
26. Addition and Subtraction of Rational Expressions
3x
4y
3x
4y
b. Convert into an expression with denominator 12xy2.
27. Addition and Subtraction of Rational Expressions
3x
4y
*12xy23x
4y =
3x
4y 12xy2
new numerator
b. Convert into an expression with denominator 12xy2.
28. Addition and Subtraction of Rational Expressions
3x
4y
*12xy23x
4y =
3x
4y 12xy2
3xy
b. Convert into an expression with denominator 12xy2.
29. Addition and Subtraction of Rational Expressions
3x
4y
*12xy23x
4y =
3x
4y 12xy2 =
9x2y
12xy2
3xy
b. Convert into an expression with denominator 12xy2.
30. Addition and Subtraction of Rational Expressions
3x
4y
*12xy2
x + 1
2x + 3
3x
4y =
3x
4y 12xy2 =
9x2y
12xy2
3xy
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.
31. Addition and Subtraction of Rational Expressions
3x
4y
*12xy2
x + 1
2x + 3
x + 1
2x + 3
3x
4y =
3x
4y 12xy2 =
9x2y
12xy2
3xy
=
x + 1
2x + 3
* (4x2 – 9) (4x2 – 9)
new numerator
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.
32. Addition and Subtraction of Rational Expressions
3x
4y
*12xy2
x + 1
2x + 3
x + 1
2x + 3
3x
4y =
3x
4y 12xy2 =
9x2y
12xy2
3xy
=
x + 1
2x + 3
* (4x2 – 9) (4x2 – 9)
=
x + 1
2x + 3
* (2x + 3)(2x – 3) (4x2 – 9)
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.
33. Addition and Subtraction of Rational Expressions
3x
4y
*12xy2
x + 1
2x + 3
x + 1
2x + 3
3x
4y =
3x
4y 12xy2 =
9x2y
12xy2
3xy
=
x + 1
2x + 3
* (4x2 – 9) (4x2 – 9)
=
x + 1
2x + 3
* (2x + 3)(2x – 3) (4x2 – 9)
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.
34. Addition and Subtraction of Rational Expressions
3x
4y
*12xy2
x + 1
2x + 3
x + 1
2x + 3
3x
4y =
3x
4y 12xy2 =
9x2y
12xy2
3xy
=
x + 1
2x + 3
* (4x2 – 9) (4x2 – 9)
=
x + 1
2x + 3
* (2x + 3)(2x – 3) (4x2 – 9)
= (x + 1)(2x – 3) (4x2 – 9)
b. Convert into an expression with denominator 12xy2.
c. Convert into an expression denominator 4x2 – 9.
35. Addition and Subtraction of Rational Expressions
3x
4y
*12xy2
c. Convert into an expression denominator 4x2 – 9.
x + 1
2x + 3
x + 1
2x + 3
3x
4y =
3x
4y 12xy2 =
9x2y
12xy2
3xy
=
x + 1
2x + 3
* (4x2 – 9) (4x2 – 9)
=
x + 1
2x + 3
* (2x + 3)(2x – 3) (4x2 – 9)
= (x + 1)(2x – 3) (4x2 – 9)
=
2x2 – x – 3
4x2 – 9
b. Convert into an expression with denominator 12xy2.
36. Addition and Subtraction of Rational Expressions
We give two methods of combining rational expressions below.
37. Addition and Subtraction of Rational Expressions
We give two methods of combining rational expressions below.
The first one is the traditional method.
38. Addition and Subtraction of Rational Expressions
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
39. Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
40. Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
41. Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
42. Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
III. Add or subtract the new numerators.
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
43. Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
III. Add or subtract the new numerators.
IV. Simplify the result.
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
44. Example C. Combine
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
III. Add or subtract the new numerators.
IV. Simplify the result.
2
3xy
–
x
2y2
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
45. Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
III. Add or subtract the new numerators.
IV. Simplify the result.
2
3xy
–
x
2y2
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
46. Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
III. Add or subtract the new numerators.
IV. Simplify the result.
2
3xy
–
x
2y2
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
47. Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
III. Add or subtract the new numerators.
IV. Simplify the result.
2
3xy
–
x
2y2
2
3xy = 6xy2
x
2y2 =
3x2
6xy2
4y
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
48. Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
III. Add or subtract the new numerators.
IV. Simplify the result.
2
3xy
–
x
2y2
2
3xy = 6xy2
x
2y2 =
3x2
6xy2
4y
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
49. Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
III. Add or subtract the new numerators.
IV. Simplify the result.
2
3xy
–
x
2y2
2
3xy = 6xy2
x
2y2 =
3x2
6xy2
2
3xy –
x
2y2 =
4y
6xy2 –
3x2
6xy2Hence
4y
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
50. Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
III. Add or subtract the new numerators.
IV. Simplify the result.
2
3xy
–
x
2y2
2
3xy = 6xy2
x
2y2 =
3x2
6xy2
2
3xy –
x
2y2 =
4y
6xy2 –
3x2
6xy2 =Hence
4y – 3x2
6xy2
4y
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
51. Example C. Combine
The LCM of the denominators {3xy, 2y2} is 6xy2.
Convert
Addition and Subtraction of Rational Expressions
Traditional Method (Optional)
(Combining fractions with different denominators)
I. Find the LCD of the expressions.
II. Convert each expression into the LCD.
III. Add or subtract the new numerators.
IV. Simplify the result.
2
3xy
–
x
2y2
2
3xy = 6xy2
x
2y2 =
3x2
6xy2
2
3xy –
x
2y2 =
4y
6xy2 –
3x2
6xy2 =Hence
4y – 3x2
6xy2
This is simplified because the numerator is not factorable.
4y
We give two methods of combining rational expressions below.
The first one is the traditional method. The second one uses
the Multiplier Method directly.
53. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 =
2x2 + x – 2 =
54. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 =
55. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
56. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
57. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
58. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2
=
x
2(2x – 1)
59. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2
=
x
2(2x – 1)
* 2(2x – 1)(x + 1) LCD
60. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2
=
x
2(2x – 1)
* 2(2x – 1)(x + 1) LCD
61. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2
=
x
2(2x – 1)
* 2(2x – 1)(x + 1) LCD
= x(x + 1) LCD
62. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2
=
x
2(2x – 1)
* 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
63. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2
=
x
2(2x – 1)
* 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
x – 1
2x2 + x – 1 = x – 1
(2x – 1)(x + 1)
64. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2
=
x
2(2x – 1)
* 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
x – 1
2x2 + x – 1 = x – 1
(2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD
65. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2
=
x
2(2x – 1)
* 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
x – 1
2x2 + x – 1 = x – 1
(2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD
66. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2
=
x
2(2x – 1)
* 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
x – 1
2x2 + x – 1 = x – 1
(2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD
= 2(x – 1) LCD
67. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2
=
x
2(2x – 1)
* 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
x – 1
2x2 + x – 1 = x – 1
(2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD
= 2(x – 1) =
2x – 2
LCD LCD
68. Example D. Combine
Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
Factor each denominator to find the LCD.
4x – 2 = 2(2x – 1),
2x2 + x – 2 = (2x – 1)(x + 1)
Hence the LCD = 2(2x – 1)(x + 1)
Next, convert each fraction into the LCD
x
4x – 2
=
x
2(2x – 1)
* 2(2x – 1)(x + 1) LCD
= x(x + 1) =
x2 + x
LCD
LCD
x – 1
2x2 + x – 1 = x – 1
(2x – 1)(x + 1) * 2(2x – 1)(x + 1) LCD
= 2(x – 1) =
2x – 2
LCD LCD
Hence x
4x – 2 – x – 1
2x2 + x – 1 = x2 + x
LCD – 2x – 2
LCD
69. Addition and Subtraction of Rational Expressions
x
4x – 2
–
x – 1
2x2 + x – 1
= x2 + x
LCD – 2x – 2
LCD
70. Addition and Subtraction of Rational Expressions
=
x2 + x – (2x – 2)
LCD
x
4x – 2
–
x – 1
2x2 + x – 1
= x2 + x
LCD – 2x – 2
LCD
71. Addition and Subtraction of Rational Expressions
=
x2 + x – (2x – 2)
LCD
x
4x – 2
–
x – 1
2x2 + x – 1
= x2 + x
LCD – 2x – 2
LCD
= x2 + x – 2x + 2
LCD
72. Addition and Subtraction of Rational Expressions
=
x2 + x – (2x – 2)
LCD
x
4x – 2
–
x – 1
2x2 + x – 1
= x2 + x
LCD – 2x – 2
LCD
= x2 + x – 2x + 2
LCD
=
x2 – x + 2
2(2x – 1)(x + 1)
73. Addition and Subtraction of Rational Expressions
=
x2 + x – (2x – 2)
LCD
x
4x – 2
–
x – 1
2x2 + x – 1
= x2 + x
LCD – 2x – 2
LCD
= x2 + x – 2x + 2
LCD
=
x2 – x + 2
2(2x – 1)(x + 1)
This is simplified because the numerator is not factorable.
74. Addition and Subtraction of Rational Expressions
=
x2 + x – (2x – 2)
LCD
x
4x – 2
–
x – 1
2x2 + x – 1
= x2 + x
LCD – 2x – 2
LCD
= x2 + x – 2x + 2
LCD
=
x2 – x + 2
2(2x – 1)(x + 1)
This is simplified because the numerator is not factorable.
Multiplier Method
75. Addition and Subtraction of Rational Expressions
=
x2 + x – (2x – 2)
LCD
x
4x – 2
–
x – 1
2x2 + x – 1
= x2 + x
LCD – 2x – 2
LCD
= x2 + x – 2x + 2
LCD
=
x2 – x + 2
2(2x – 1)(x + 1)
This is simplified because the numerator is not factorable.
Multiplier Method
The Multiplier Method is the same method used for fractional
numbers.
76. Addition and Subtraction of Rational Expressions
=
x2 + x – (2x – 2)
LCD
x
4x – 2
–
x – 1
2x2 + x – 1
= x2 + x
LCD – 2x – 2
LCD
= x2 + x – 2x + 2
LCD
=
x2 – x + 2
2(2x – 1)(x + 1)
This is simplified because the numerator is not factorable.
Multiplier Method
The Multiplier Method is the same method used for fractional
numbers. We used the fact that x = (x * LCD) / LCD = x * 1
77. Addition and Subtraction of Rational Expressions
=
x2 + x – (2x – 2)
LCD
x
4x – 2
–
x – 1
2x2 + x – 1
= x2 + x
LCD – 2x – 2
LCD
= x2 + x – 2x + 2
LCD
=
x2 – x + 2
2(2x – 1)(x + 1)
This is simplified because the numerator is not factorable.
Multiplier Method
The Multiplier Method is the same method used for fractional
numbers. We used the fact that x = (x * LCD) / LCD = x * 1
then compute (x * LCD) using the Distributive Law.
78. Addition and Subtraction of Rational Expressions
=
x2 + x – (2x – 2)
LCD
x
4x – 2
–
x – 1
2x2 + x – 1
= x2 + x
LCD – 2x – 2
LCD
= x2 + x – 2x + 2
LCD
=
x2 – x + 2
2(2x – 1)(x + 1)
This is simplified because the numerator is not factorable.
Multiplier Method
The Multiplier Method is the same method used for fractional
numbers. We used the fact that x = (x * LCD) / LCD = x * 1
then compute (x * LCD) using the Distributive Law. Following
is an example using this method.
81. Example E. Calculate
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
82. Example E. Calculate
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )
83. Example E. Calculate
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72
84. Example E. Calculate
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
85. Example E. Calculate
6
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
86. Example E. Calculate
6 9
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
87. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
88. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
89. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
90. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. Combine 3
4xy2
– 5x
6y
91. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
92. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
93. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2)
94. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2) Distribute
95. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2) Distribute
3
96. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2) Distribute
3 2xy
97. Example E. Calculate
6 9 8
Addition and Subtraction of Rational Expressions
7
12
+
5
8
–
4
9
The LCD is 72. Multiply the problem by the LCD then put the
result over the new denominator, the LCD.
7
12
+
5
8
–
4
9
( )* 72 72 Distribute the multiplication
= ( 42 + 45 – 32 ) 72
55
72
=
Example F. Combine 3
4xy2
– 5x
6y
The LCD is 12 xy2.
Multiply then divide the problem by the LCD.
3
4xy2
– 5x
6y
( ) * 12xy2 / (12xy2) Distribute
3 2xy
9 – 10x2y
12xy2=
98. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4).
99. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
100. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4)
101. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
102. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
103. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
2x + 26
(x – 2)(x + 4)
= 2(x + 13)
(x – 2)(x + 4)
or
104. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
2x + 26
(x – 2)(x + 4)
=
Example H. Combine x
x2 – 2x
– x – 1
x2 – 4
2(x + 13)
(x – 2)(x + 4)
or
105. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
2x + 26
(x – 2)(x + 4)
=
Example H. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
2(x + 13)
(x – 2)(x + 4)
or
106. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
2x + 26
(x – 2)(x + 4)
=
Example H. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
x2 – 2x = x(x – 2)
2(x + 13)
(x – 2)(x + 4)
or
107. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
2x + 26
(x – 2)(x + 4)
=
Example H. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
x2 – 2x = x(x – 2)
x2 – 4 = (x – 2)(x + 2)
2(x + 13)
(x – 2)(x + 4)
or
108. Addition and Subtraction of Rational Expressions
Example G. Combine 5
x– 2
– 3
x + 4
The LCD is (x – 2)(x + 4). Hence
= [5(x + 4) – 3(x – 2)] / (x – 2)(x + 4)
5
x– 2
– 3
x + 4
( ) (x – 2)(x + 4) / (x – 2)(x + 4)
(x + 4) (x – 2)
2x + 26
(x – 2)(x + 4)
=
Example H. Combine x
x2 – 2x
– x – 1
x2 – 4
Factor each denominator to find the LCD.
x2 – 2x = x(x – 2)
x2 – 4 = (x – 2)(x + 2)
Hence the LCD = x(x – 2)(x + 2).
2(x + 13)
(x – 2)(x + 4)
or
109. Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
110. Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
=
x
x2 – 2x
– x – 1
x2 – 4
111. * x( x – 2)(x + 2)
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4
112. * x( x – 2)(x + 2)
(x + 2)
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4
113. * x( x – 2)(x + 2)
(x + 2) x
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4
114. * x( x – 2)(x + 2)
(x + 2) x
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4
= [x(x + 2) – x(x – 1)] LCD
115. * x( x – 2)(x + 2)
(x + 2) x
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4
= [x(x + 2) – x(x – 1)] LCD
= [x2 + 2x – x2 + x)] LCD
116. * x( x – 2)(x + 2)
(x + 2) x
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4
= [x(x + 2) – x(x – 1)] LCD
= [x2 + 2x – x2 + x)] LCD
= 3x
x (x – 2)(x + 2)
117. * x( x – 2)(x + 2)
(x + 2) x
Multiply the problem by the LCD then put the result over the
new denominator, the LCD.
Addition and Subtraction of Rational Expressions
x
x(x – 2)
– (x – 1)
(x – 2)(x + 2)
[ ] LCD=
x
x2 – 2x
– x – 1
x2 – 4
= [x(x + 2) – x(x – 1)] LCD
= [x2 + 2x – x2 + x)] LCD
= 3x
x (x – 2)(x + 2)
=
3
(x – 2)(x + 2)
118. Ex. A. Combine and simplify the answers.
Addition and Subtraction of Rational Expressions
x
x – 2
– 2
x – 2
1.
2x
x – 2
+
4
x – 2
2.
3x
x + 3
+ 6
x + 3
3.
– 2x
x – 4
+
8
x – 4
4.
x + 2
2x – 1
–
2x – 1
5.
2x + 5
x – 2
–
4 – 3x
2 – x
6.
x2 – 2
x – 2
– x
x – 2
7. 9x2
3x – 2
– 4
3x – 2
8.
Ex. B. Combine and simplify the answers.
3
12
+
5
6
–
2
3
9.
11
12+
5
8
–
7
6
10.
–5
6
+
3
8
– 311.
12.
6
5xy2
– x
6y13.
3
4xy2
– 5x
6y
15.
7
12xy
– 5x
8y316.
5
4xy
– 7x
6y214.
3
4xy2
– 5y
12x217.
–5
6 –
7
12+ 2
+ 1 – 7x
9y2
4 – 3x