5: Rational Expressions and Equations
- Page ID
- 142789
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 5.1: Simplify Rational Expressions
- This page covers handling rational expressions, focusing on identifying undefined values, evaluating expressions, and simplifying through common factors. Key concepts include avoiding division by zero, defining simplified forms, and employing techniques like factoring. The content provides definitions, examples, and practice problems for clarity, emphasizing the significance of recognizing opposite factors and ensuring expressions are fully simplified.
- 5.2: Multiply and Divide Rational Expressions
- This page discusses the multiplication and division of rational expressions, outlining methods for both operations. For multiplication, the process involves multiplying numerators and denominators while simplifying by removing common factors. Division is approached by converting it into multiplication using the reciprocal of the second expression, with an emphasis on factoring and simplification.
- 5.3: Add and Subtract Rational Expressions with a Common Denominator
- This page explains how to add and subtract rational expressions with a common denominator. It emphasizes matching denominators, simplifying results, and checking for values that could make the denominator zero. The page provides definitions and examples of both operations, highlighting the importance of combining like terms and factoring. It outlines step-by-step solutions to clarify the processes involved in manipulations while maintaining a common denominator.
- 5.4: Add and Subtract Rational Expressions with Unlike Denominators
- This page teaches how to add and subtract rational expressions by finding the least common denominator (LCD), factoring expressions, and rewriting them accordingly. It emphasizes the significance of simplifying results and careful handling of signs during operations. The text provides step-by-step guidance through examples and encourages identifying common factors for effective simplification.
- 5.5: Simplify Complex Rational Expressions
- This page explains how to simplify complex rational expressions through two methods: rewriting them as division problems and utilizing the least common denominator (LCD). It includes multiple examples demonstrating the step-by-step approach of finding the LCD, multiplying by it, and simplifying. The section emphasizes avoiding values that make denominators zero and provides readiness quizzes to reinforce learning.
- 5.6: Solve Rational Equations
- This page provides a comprehensive guide on solving rational equations and expressions, highlighting the significance of avoiding extraneous solutions that render equations undefined. It outlines key strategies, including finding the least common denominator, clearing fractions, and using techniques like combining like terms and factoring. Through multiple examples, it illustrates problem-solving methods while ensuring the verification of solutions and addressing cases with no valid solutions.
- 5.7: Solve Uniform Motion and Work Applications
- This page discusses problem-solving in uniform motion and work applications, providing examples such as travel times for trains and buses and computing speeds for various activities. Key concepts include the D=rt formula and equation setup for distance and speed, emphasizing data organization and answer verification. It explores different scenarios, including biking speeds and collaborative tasks like painting and gardening, demonstrating how teamwork enhances efficiency.