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9Th Grade | Unit 6 MATH STUDENT BOOK 9th Grade | Unit 6 804 N. 2nd Ave. E. Rock Rapids, IA 51246-1759 800-622-3070 www.aop.com Unit 6 | Algebraic Fractions Math 906 Algebraic Fractions INTRODUCTION |3 1. OPERATIONS 5 REDUCING FRACTIONS |6 ADDING AND SUBTRACTING FRACTIONS |14 MULTIPLYING AND DIVIDING FRACTIONS |22 SIMPLIFYING COMPLEX FRACTIONS |29 SELF TEST 1 |34 2. OPEN SENTENCES 37 SOLVING EQUATIONS |37 SOLVING INEQUALITIES |47 REWRITING FORMULAS |53 SELF TEST 2 |58 3. WORD PROBLEMS 61 WORK AND SINGLE-NUMBER |62 MOTION AND CONSECUTIVE-NUMBER |67 AGE AND QUOTIENT-REMAINDER |71 RATIO AND MIXTURE |77 SELF TEST 3 |86 GLOSSARY |92 LIFEPAC Test is located in the center of the booklet. Please remove before starting the unit. Section 1 |1 Algebraic Fractions | Unit 6 Author: Arthur C. Landrey, M.A.Ed. Editor-In-Chief: Richard W. Wheeler, M.A.Ed. Editor: Robin Hintze Kreutzberg, M.B.A. Consulting Editor: Robert L. Zenor, M.A., M.S. Revision Editor: Alan Christopherson, M.S. Westover Studios Design Team: Phillip Pettet, Creative Lead Teresa Davis, DTP Lead Nick Castro Andi Graham Jerry Wingo 804 N. 2nd Ave. E. Rock Rapids, IA 51246-1759 © MCMXCVI by Alpha Omega Publications, Inc. All rights reserved. LIFEPAC is a registered trademark of Alpha Omega Publications, Inc. All trademarks and/or service marks referenced in this material are the property of their respective owners. Alpha Omega Publications, Inc. makes no claim of ownership to any trademarks and/ or service marks other than their own and their affiliates, and makes no claim of affiliation to any companies whose trademarks may be listed in this material, other than their own. 2| Section 1 Unit 6 | Algebraic Fractions Algebraic Fractions INTRODUCTION In this LIFEPAC® you will continue your study in algebra. You will apply what you have learned so far to fractions having polynomial numerators or denominators or both. The factoring techniques that you learned in Mathematics LIFEPAC 905 will be used when performing the basic operations with these fractions. Then you will solve open sentences containing fractions by methods that are quite similar to those you have already used. Finally you will have another opportunity to solve verbal problems, this time in applications that involve fractions. Objectives Read these objectives. The objectives tell you what you will be able to do when you have successfully completed this LIFEPAC. When you have finished this LIFEPAC, you should be able to: 1. Determine the excluded value(s) for a fraction. 2. Reduce a fraction to lowest terms. 3. Find sums and differences of fractions. 4. Find products and quotients of fractions. 5. Simplify complex fractions. 6. Solve equations containing fractions. 7. Solve inequalities containing fractions. 8. Change the subject of a formula containing fractions. 9. Solve problems requiring the use of fractions. Section 1 |3 Algebraic Fractions | Unit 6 Survey the LIFEPAC. Ask yourself some questions about this study and write your questions here. ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ ________________________________________________________________________________________________ 4| Section 1 Unit 6 | Algebraic Fractions 1. OPERATIONS As you work through this first section, keep We will begin by defining algebraic in mind that the basic concepts of reducing, fractions, since we must know what they adding, subtracting, multiplying, dividing, are in order to be able to work with them. and simplifying the fractions of algebra are the same as those used for the fractions of arithmetic. OBJECTIVES Review these objectives. When you have completed this section, you should be able to: 1. Determine the excluded value(s) for a fraction. 2. Reduce a fraction to lowest terms. 3. Find sums and differences of fractions. 4. Find products and quotients of fractions. 5. Simplify complex fractions. VOCABULARY A Algebraic fraction—an indicated quotient of two polynomials written in the form B . A is the numerator of the algebraic fraction and B is the denominator. Terms—the numerator and denominator of a fraction. 2 -y2 – 3y + 1 a + b + c k – 3 Models: x + 3 5 – 2y m – n 7 Algebraic fractions can be reduced, multiplication, division, and simplification using similar methods as for reducing are also possible with algebraic fractions. arithmetic fractions. Addition, subtraction, Section 1 |5 Algebraic Fractions | Unit 6 REDUCING FRACTIONS Algebraic fractions can be reduced by the denominator of the fourth fraction is finding the lowest terms. First, however, the constant 7 and 7 ≠ 0, that fraction has we need to discuss the circumstances no excluded values. under which algebraic fractions may not even exist! In determining the excluded values for x – 3 the fraction x2 – 4 , you may be able to see 2 EXCLUDED VALUES immediately that 2 – 4 = 0; thus, x = 2 is A an excluded value. However, (-2)2 – 4 = 0 is Since a fraction indicates division ( = A ÷ B also true; thus, x = -2 is an excluded value B) and since division by zero is undefined, as well. the denominator of a fraction must be nonzero (B ≠ 0). If a denominator contains In Mathematics LIFEPAC 905 you learned any variables, then a value that would to factor, and now factoring can be used to result in zero for that denominator must be find both these excluded values. Since the excluded for the fraction to exist. denominator x2 – 4 is a difference of two squares, it has factors (x + 2)(x – 2). The In the preceding models, the denominators first factor, x + 2, would become zero if are x + 3, 5 – 2y, m – n, and 7, x = -2; likewise, the second factor, x – 2, respectively. The excluded values are x = would become zero if x = 2. The excluded -3 for the first model (-3 + 3 = 0) ,y = 2.5 values are then x = 2 and x = -2. In this for the second (5 – 2 • 2.5 = 0), and m = method we have made use of an important n for the third (m – m or n – n = 0); since property in mathematics. Property If A • B = 0, then A = 0 or B = 0 (or both); if a product of factors is zero, then at least one of the factors must be zero. 6| Section 1 Unit 6 | Algebraic Fractions Model 1: Find the excluded value(s) for the fraction a + 5 . a(b + 3)(c – 2) Solution: The denominator is already factored, so each of the three factors is set equal to zero. a = 0 b + 3 = 0 c – 2 = 0 b = -3 c = 2 ∴ The excluded values are a = 0, b = -3, and c = 2. 7 Model 2: Find the excluded value(s) for the fraction 2 . d – 5d – 24 Solution: The factors of d 2 – 5d – 24 are (d – 8)(d + 3). d – 8 = 0 d + 3 = 0 d = 8 d = -3 ∴ The excluded values are d = 8 and d = -3. Check: If d = 8, d 2 – 5d – 24 = (8)2 – 5 (8) – 24 = 64 – 40 – 24 = 0. If d = -3, d 2 – 5d – 24 = (-3)2 – 5(-3) – 24 = 9 + 15 – 24 = 0. Write the excluded value(s) for each fraction, or none if that is the case. 1.1 a 1.6 a b – 2 ______________ 3 – 2a ______________ 1.2 4x + 3 1.7 x + 3 x ______________ y(z + 5) ______________ 2 2 1.3 y – y + 5 1.8 k + 5k + 1 y + 4 ______________ k2 – 9 ______________ 3 1.4 3 1.9 7b 5n ______________ b2 – 7b + 10 ______________ 1.5 -2x 1.10 x + 11 17 ______________ 3x2 + 5x – 2 ______________ Section 1 |7 Algebraic Fractions | Unit 6 As you work through this LIFEPAC, you LOWEST TERMS are to assume that all fractions do exist; Now you are ready to begin working with that is, any value(s) that would make a these algebraic fractions. A basic denominator zero are understood to be property of fractions will be used in much excluded. However, from time to time of this work. (as in the preceding activities), you will be asked to identify these excluded values. Property A = A C (or A C = A ) for C ≠ 0 B BC BC B ; if the numerator and the denominator of a fraction are both multiplied (or divided) by the same nonzero value, then an equivalent fraction is obtained. In arithmetic you learned that the 2 1 the fraction 3 since both the numerator 12 fraction 2 has the same value as the 5 and the denominator
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