module05_MAT-115-sep13 - MAT115:INTERMEDIATEALGEBRA...

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Unformatted text preview: MAT­115: INTERMEDIATE ALGEBRA Module 5—Rational Expressions OBJECTIVES After successfully completing Module 5, you should be able to: ● Use the basic principle of rational numbers to change the form of rational expressions. ● Multiply and divide rational expressions. ● Add and subtract rational expressions. ● Simplify complex algebraic fractions. ● Apply equation­solving strategies to solving equations with rational expressions. ● Recognize extraneous solutions. ● Solve applications involving equations with rational expressions. ● Understand the relationship between a ratio and a proportion and solve applications involving them. STUDY MATERIALS Reading and Viewing Assignment ● Textbook:​ Sections 6.1 through 6.4 and 6.6 through 6.7 (Lesson 13) ● Telecourse Student Guide:​ Lesson 13 ● Video Program: ○ Rational Expressions Study Notes ● Section 6.1 (Lesson 13) ○ If an algebraic expression is fractional in form (i.e., it has a polynomial in the numerator or the denominator or both), it is called a ​ rational expression​ . ○ The basic principle of rational numbers (fractions) applies to rational expres​ sions when reducing to lower terms or raising to higher terms. ○ When reducing to lowest terms, ​ only common factors in products may be divided out​ . A common mistake is dividing out common terms in sums or differences. Use the "Strategy for Reducing Rational Expressions" on page 385 to avoid this error when simplifying rational expressions. Copyright © 2013 by Thomas Edison State College. All rights reserved. ○ To "simplify" a rational expression means to reduce it to lowest terms. ○ The principle of raising fractions to higher terms is important when adding and subtracting rational expressions. ● Section 6.2 (Lesson 13) ○ When multiplying rational expressions, always factor where possible and note that common factors can be divided out. ○ Division of rational expressions requires multiplication by the reciprocal of the divisor; therefore, always rewrite a division problem as a product (i.e., the product of the dividend multi​ plied by the reciprocal of the divisor). ○ The denominator of a rational expression cannot equal zero. ○ A rational expression may have the same terms in the numera​ tor and denominator but with opposite signs. For example: (​ a​ – ​ b​ ) / (​ b​ – ​ a​ ). To simplify such a rational expression requires "dividing out opposites." Examples 2(b) and 3 (pp. 389–390) illustrate methods for simplifying such expressions. ● Section 6.3 (Lesson 13) ○ Rational expressions must have the same (common) denomi​ nator if we are to add or subtract them. ○ If the denominators are different, then we need to find a least common denominator (LCD) before adding or subtracting the rational expressions. To accomplish this, first find the least common multiple (LCM) of all the denominators by factoring each denominator completely. Convert each term to a fraction with the LCD as the denominator, and apply the same factor to the numerator. Always express your answer in lowest terms. ● Section 6.4 (Lesson 13) ○ Complex fractions are expressions with fractions in the numerator or the denominator. ○ To simplify complex fractions, you may use the same proce​ dure discussed in Section 6.2 for dividing rational expressions. ○ Complex fractions may also be simplified by finding the LCD for all the rational expressions in the complex fraction, then multiplying both the numerator and denominator by the LCD to obtain a simple fraction. ● Section 6.6 (Lesson 13) ○ Recall that if an equation involves a rational expression, the first step in the solution is to Copyright © 2013 by Thomas Edison State College. All rights reserved. use multiplication to remove the fraction. The same procedure is used if the equation has a variable in one or more denominators. ○ See the discussion on page 430 about "extraneous roots" and the importance of checking proposed solutions in original equations. The denominator of a rational expression must not equal zero. Determine the domain of the variable before solving the equation to recognize extraneous roots immediately. ○ A ratio expresses a relationship between two quantities and may be written as a fraction x​ /​ y​ , where ​ y​ 0, or as ​ a​ : ​ b​ . ○ An unknown quantity may be determined if the ratio between quantities is known. Two ratios are set equal to each other and solved for the unknown quantity. The extremes­means property (cross multiplication) may some​ times be used in solving such a problem. See the "Caution" on page 432 about when it is appropri​ ate to use this property. ○ A proportion expresses the equality between two ratios. ○ Only quantities that are directly proportional are covered in this section. Other relationships may exist (e.g., indirect proportion​ ality). ● Section 6.7 (Lesson 13) ○ The real­world applications in Section 6.7 involve solving formulas and problems stated in words that require translation into algebraic equations. For a summary of "Verbal Phrases and Algebraic Expressions" and a "Strategy for Solving Word Problems," see Section 2.3, pages 90 and 91. Self­Tests The textbook publisher has provided self­tests for each chapter and section of the Dugopolski text. Click here​ to launch the self­tests. ACTIVITIES Module 5 has two activities. Please consult the course Calendar for the due dates. Group Activity 5 Phase 1: Group Work Phase 1 of the Group Activity involves group problem solving and discussion based on textbook exercises similar to those on the written assignment. Participation in group work entails posting draft solutions to Copyright © 2013 by Thomas Edison State College. All rights reserved. selected exercises and discussing them with other group members. "Group work" counts 60 percent toward your activity grade. Here's how to proceed: 1. Select two exercises from each of the two section groupings listed below, for a total of ​ four ​ (4) exercises. Post a message to the Group Work Forum indicating which exercises you have chosen so that other members of your group will know to select other exercises. A. Section 6.1: ​ exercises 2, 18, 48, 64, 84 Section 6.2: ​ exercises 10, 28, 38, 78, 86 Section 6.3:​ exercises 2, 26, 44, 74, 82 B. Section 6.4: ​ exercises 2, 18, 26, 48, 52 Section 6.6:​ exercises 2, 6, 26, 54, 56 Section 6.7:​ exercises 2, 18 Note:​ Exercises are selected on a first­come basis. If a member of your group has already chosen an exercise you had wanted to answer, please select a different exercise. Your goal as a group is to answer and discuss as many different exercises as possible. Also, don't shy away from the seemingly more difficult exercises in each set. Some of these will help prepare you for similar questions on the module's written assignment. 2. Post your draft solutions to the Group Work Forum. Be sure to identify the section and exercise numbers and to type out each question. 3. Discuss your work freely with other members of the group. Share ideas and advice, ask questions, and help each other out as much as possible. Ultimately, however, you will be responsible for posting your own solutions to the Solutions Forum (see phase 2). Phase 2: Posting Your Solutions Phase 2 of the Group Activity involves posting worked­out solutions to the four exercises you selected in phase 1. You post these solutions to the Solutions Forum. Your worked­out solutions to four exercises count 40 percent toward your activity grade. Partial credit will be given. Here's how to proceed: 1. In Solutions Forum 5, post your worked­out solutions to the four exercises you selected in phase 1 of the Group Activity. Please title your posting "Module 5 Exercises/[Your Name]." 2. Read and comment on the solutions posted by your classmates. Your mentor will similarly review and comment on the solutions provided. Study the mentor's comments closely. Written Assignment 5 The written assignment draws on even­numbered exercises from the textbook. ​ Answer all assigned exercises, and show all work. ​ (Note: See graph pools for graphs that can be incorporated into the assignment.) Copyright © 2013 by Thomas Edison State College. All rights reserved. ● Section 6.1: ​ exercises ​ 10, 20, 56, 66, 78, 88 ● Section 6.2: ​ exercises 14, 22, ​ 30, 40, 72, 76, 82 ● Section 6.3:​ exercises ​ 6, 16, 20, 36, 46, 70, 100 ● Section 6.4: ​ exercises ​ 6, 20, 22, 28, 42, 58, 60, 66 ● Section 6.6: ​ exercises ​ 10, 20, 32, 50, 58, 82 ● Section 6.7:​ exercises ​ 8, 10, 20, 30, 38, 44 Assignments must be prepared electronically, using a word processor and whatever equation editor comes with your word processing software. However, if your word processor is not compatible with your mentor's word processor, you will need to save your document as a rich­text file (.rtf) before submitting it. Check with your mentor first to determine file compatibility. When preparing your answers, please identify each exercise clearly by textbook section and exercise number. To receive full credit for your answers, you must show all work and include complete solutions. Copyright © 2013 by Thomas Edison State College. All rights reserved. ...
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