Rational Expressions and Rational Equations


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1 Rational Epressions and Rational Equations 6 6. Rational Epressions and Rational Functions 6. Multiplication and Division of Rational Epressions 6. Addition and Subtraction of Rational Epressions 6.4 Comple Fractions Problem Recognition Eercises Operations on Rational Epressions 6. Rational Equations 6.6 Applications of Rational Equations and Proportions In this chapter, we define a rational epression as a ratio of two polynomials. First we focus on adding, subtracting, multiplying, and dividing rational epressions. Then the chapter concludes with solving rational equations and showing how they are used in applications. As you work through the chapter, pay attention to key terms. Then work through this crossword puzzle. Across. Epression of the form a/b. Two triangles in which corresponding sides are proportional 7. Equation of the form a/b = c/d 4 Down. Fraction with fractions in numerator and denominator. Ratio of two polynomials 4. Set of real numbers for which an epression is defined 6. Abbreviation for least common denominator
2 404 Chapter 6 Rational Epressions and Rational Equations Section 6. Concepts. Evaluating Rational Epressions. Domain of a Rational Epression. Simplifying Rational Epressions to Lowest Terms 4. Simplifying Ratios of. Rational Functions Rational Epressions and Rational Functions. Evaluating Rational Epressions In Chapter, we introduced polynomials and polynomial functions. In this chapter, polynomials will be used to define rational epressions and rational functions. Definition of a Rational Epression p An epression is a rational epression if it can be written in the form q, p and q are polynomials and q 0. where A rational epression is the ratio of two polynomials. The following epressions are eamples of rational epressions. y c,,, y 4y c 4 a a 0 Eample Evaluating a Rational Epression Evaluate the epression for the given values of. a. b. c. 0 d. e. Solution: a. b. c d. e. (undefined) 0 Skill Practice. Evaluate the epression for the given values of y. y y 4 a. y b. y 0 c. y 0 d. y 4 e. y Skill Practice Answers a. b. c. 0 d. Undefined e.. Domain of a Rational Epression The domain of an epression is the set of values that when substituted into the epression produces a real number. Therefore, the domain of a rational epression must eclude the values that make the denominator zero. For eample: Given, the domain is the set of all real numbers,, ecluding. This can be written in setbuilder notation as: 0 is a real number and 6
3 Section 6. Rational Epressions and Rational Functions 40 Steps to Find the Domain of a Rational Epression. Set the denominator equal to zero and solve the resulting equation.. The domain is the set of all real numbers ecluding the values found in step. Eample Finding the Domain of a Rational Epression Write the domain in setbuilder notation. 4 a. b. c. 4t Solution: a. The epression is undefined when the denominator is zero. 4t 4t 0 4t 4 Set the denominator equal to zero and solve for t. t 4 The value t must be ecluded from the domain. 4 Domain: t 0 t is a real number and t 46 b or Set the denominator equal to zero and solve for. This is a factorable quadratic equation. 0 or Domain: 0 is a real number and, 6 TIP: The domain of a rational epression ecludes values for which the denominator is zero. The values for which the numerator is zero do not affect the domain of the epression. c. 4 Because the quantity is nonnegative for any real number, the denominator 4 cannot equal zero; therefore, no real numbers are ecluded from the domain. Domain: 0 is a real number6
4 406 Chapter 6 Rational Epressions and Rational Equations Skill Practice Write the domain in setbuilder notation v 9 t t. Simplifying Rational Epressions to Lowest Terms As with fractions, it is often advantageous to simplify rational epressions to lowest terms. The method for simplifying rational epressions to lowest terms mirrors the process to simplify fractions. In each case, factor the numerator and denominator. Common factors in the numerator and denominator form a ratio of and can be simplified. Simplifying a fraction: Simplifying a rational epression: 6 factor factor This process is stated formally as the fundamental principle of rational epressions. Fundamental Principle of Rational Epressions Let p, q, and r represent polynomials. Then pr qr p for q 0 and r 0 q The first eample involves reducing a quotient of two monomials. This was also covered in Section.8. Eample Simplify the rational epression. Simplifying a Rational Epression to Lowest Terms y 8 4 y Solution: y 8 4 y y 4 y y y y This epression has the restriction that 0 and y 0. Factor the denominator. Reduce the common factors whose ratio is. Skill Practice Answers. v 0 v is a real number and v is a real number and, 6 4. t 0 t is a real number6 y 4 provided that 0 and y 0.
5 Section 6. Rational Epressions and Rational Functions 407 Skill Practice. 9a b 8a 8 b Simplify the rational epression. Eample 4 Simplifying a Rational Epression to Lowest Terms a. Factor the numerator and denominator. b. Determine the domain and write the domain in setbuilder notation. c. Simplify the epression Solution: a Factor the numerator and denominator. b. To determine the domain, set the denominator equal to zero and solve for The domain of the function is 0 is a real number and c. Simplify the ratio of common factors to. 4 Set 4 0. Note that the factor of 6 will never equal 0. provided 4 Avoiding Mistakes: The domain of a rational epression is always determined before simplifying the epression. Skill Practice 6. a a 8 a 4 Factor the numerator and the denominator to determine the domain. Then simplify the rational epression. It is important to note that the epressions and Skill Practice Answers b 0. and b 0 a, a 0 6. a a is a real number and a 7 6; a 4
6 408 Chapter 6 Rational Epressions and Rational Equations are equal for all values of that make each epression a real number. Therefore, for all values of ecept 4. (At 4, the original epression is undefined.) This is why the domain of a rational epression is always determined before the epression is simplified. The purpose of simplifying a rational epression is to create an equivalent epression that is simpler to evaluate. Consider the epression from Eample 4 in its original form and in its simplified form. If we substitute an arbitrary value of into the epression (such as ), we see that the simplified form is easier to evaluate. Substitute Eample Simplify to lowest terms Original Epression Simplifying a Rational Epression to Lowest Terms t 8 t 6t 8 Simplified Form 4 Solution: t 8 t 6t 8 TIP: t 8 is a sum of cubes. Recall: a b a ba ab b t 8 t t t 4 t t t 4 t t 4 t t t 4 t t 4 The restrictions on the domain are t and t 4. Reduce common factors whose ratio is. t t 4 t 4 provided t, t 4 Skill Practice Answers 4y 7. provided y, y y Skill Practice Simplify to lowest terms. 8y 4y 7. y y
7 Section 6. Rational Epressions and Rational Functions 409 Avoiding Mistakes: The fundamental principle of rational epressions indicates that common factors in the numerator and denominator may be simplified. pr qr p q r r p q p q Because this property is based on the identity property of multiplication, reducing applies only to factors (remember that factors are multiplied). Therefore, terms that are added or subtracted cannot be reduced. For eample: y y y y Reduce common factor. However, y cannot reduce common terms cannot be simplified. 4. Simplifying Ratios of When two factors are identical in the numerator and denominator, they form a ratio of and can be simplified. Sometimes we encounter two factors that are opposites and form a ratio of. For eample: Simplified Form Details/Notes The ratio of a number and its opposite is. The ratio of a number and its opposite is. Factor out. Recognizing factors that are opposites is useful when simplifying rational epressions. For eample, a b and b a are opposites because the opposite of a b can be written a b a b b a. Therefore, in general, a b, provided a b. b a Eample 6 Simplifying Rational Epressions Simplify the rational epressions to lowest terms. a. b. 6
8 40 Chapter 6 Rational Epressions and Rational Equations Solution: a. a b or Factor. The restrictions on are and. Notice that and are opposites and form a ratio of. In general, a b b a. provided, TIP: The factor of may be applied in front of the rational epression, or it may be applied to the numerator or to the denominator. Therefore, the final answer may be written in several forms. or or b. 6 Factor. The restriction on is. Recall: a b b a. provided Skill Practice Simplify the epressions. b a a ab b Skill Practice Answers 8. a b a b 9. or a b a b. Rational Functions Thus far in the tet, we have introduced several types of functions including constant, linear, and quadratic functions. Now we will introduce another category of functions called rational functions.
9 Section 6. Rational Epressions and Rational Functions 4 Definition of a Rational Function A function f is a rational function if it can be written in the form where p and q are polynomial functions and q 0. f p q, For eample, the functions f, g, h, and k are rational functions. f a 6, g, ha a, k 4 In Section 4., we introduced the rational function defined by f. Recall that f has a restriction on its domain that and the graph of f 0 has a vertical asymptote at 0 (Figure 6). f() 4 f() 4 4 Figure 64 To evaluate, simplify, and find the domain of rational functions, we follow procedures similar to those shown for rational epressions. Eample 7 Evaluating a Rational Function Given a. Find the function values (if they eist) g(0), g(), g(), g(.), g(.9), g(), g(.), g(.), g(4), and g(). b. Write the domain of the function in interval notation. Solution: g a. g0 g undefined 0 0 g g g g g. g g4 4 g
10 4 Chapter 6 Rational Epressions and Rational Equations b. g 0 The value of the function is undefined when the denominator equals zero. Set the denominator equal to zero and solve for. The value must be ecluded from the domain. Domain:,, 4 0 ( ( 4 Skill Practice 0. Given h find the function values. 6, a. h(0) b. h() c. h() d. h(6) e. Write the domain of h in interval notation. g() 4 g() 4 4 Figure The graph of g is shown in Figure 6. Notice that the function has a vertical asymptote at where the function is undefined (Figure 6). Calculator Connections A Table feature can be used to check the function values in Eample 7, g The graph of a rational function may be misleading on a graphing calculator. For eample, g has a vertical asymptote at, but the vertical asymptote is not part of the graph (Figure 6). A graphing calculator may try to connect the dots between two consecutive points, one to the left of and one to the right of. This creates a line that is nearly vertical and appears to be part of the graph. To show that this line is not part of the graph, we can graph the function in a Dot Mode (see the owner s manual for your calculator). The graph of g in dot mode indicates that the line is not part of the function (Figure 64). Figure 6 Figure 64 Skill Practice Answers 0a. b. c. 0 d. Undefined e., 6 6,
11 Section 6. Rational Epressions and Rational Functions 4 Section 6. Boost your GRADE at mathzone.com! Study Skills Eercise Practice Eercises Practice Problems SelfTests NetTutor. Define the key terms. a. Rational epression b. Rational function eprofessors Videos Concept : Evaluating Rational Epressions. Evaluate the epression for 0,,, Evaluate the epression for 6, 4, 0, and 4. 6 y 4. Evaluate the epression for y 0, y, y, y 4. y a. Evaluate the epression for a a, a 0, a a., z 8 6. Evaluate the epression for z 4, z 4, z, z. z 9 Concept : Domain of a Rational Epression For Eercises 7 6, determine the domain of the rational epression. Write the answer in setbuilder notation. 9 0 v t y a v 8 t 6t q t 8 q 6q 7 b b b c. 6. c d d 6 Concept : Simplifying Rational Epressions to Lowest Terms Given:, 4 a. Factor the numerator and denominator. b. Write the domain in setbuilder notation. c. Simplify the epression.
12 44 Chapter 6 Rational Epressions and Rational Equations 6 8. Given: f 6 a. Factor the numerator and denominator. b. Write the domain in setbuilder notation. c. Simplify the epression. 9. Given: p a. Factor the numerator and denominator. b. Write the domain in setbuilder notation. c. Simplify the epression. 0. Given: q a. Factor the numerator and denominator. b. Write the domain in setbuilder notation. c. Simplify the epression. For Eercises 4, simplify the rational epressions.. 00 y 48ab c 7w z 6.. 6y 8 6a 7 bc 0 4w z 4.. m 4 n y 6a m 6 n y 9a z 6 7c 0.. z c c.. t 7t 4 y 8y 9 p p t t y y 4 p p z c c z z z 8 c c 0 z z 40. r 9 s 4r 8 s 4 y y 9 p p 7p 6 r r r r p p 4 p Concept 4: Simplifying Ratios of For Eercises 4 6, simplify the rational epressions. r 6 a b r a b 8 0 y 4 t y t 7 w 7 w p 0 p
13 Section 6. Rational Epressions and Rational Functions 4 c 4 b y c 4 b y 4w 49z 4z 4w y 6 y y Mied Eercises For Eercises 7 70, simplify the rational epression. 7. t 4 4 8p t 7t 6 4 p 4p a bc 4 9 yz y ab c yz 8y b 4 c b b c 7c y y 9y p 49 4 p 9w w a a a a t t 4 t t Concept : Rational Functions 7. Let h. Find the function values h0, h, h, h, h. 7. Let k. Find the function values k 0, k, k, k, k. For Eercises 7 76, find the domain of each function and use that information to match the function with its graph. 7. m 74. n 7. q p a. b. c. d. 4 y y y y For Eercises 77 86, write the domain of the function in interval notation. 77. f 78. r t 79. t qt t p 8. w 8. st 6 4t 6
14 46 Chapter 6 Rational Epressions and Rational Equations 8. m 84. n r st t t 8t 6t Epanding Your Skills 87. Write a rational epression whose domain is,,. (Answers may vary.) 88. Write a rational epression whose domain is,,. (Answers may vary.) 89. Write a rational function whose domain is,,. (Answers may vary.) 90. Write a rational function whose domain is, 6 6,. (Answers may vary.) Section 6. Concepts. Multiplication of Rational Epressions. Division of Rational Epressions Multiplication and Division of Rational Epressions. Multiplication of Rational Epressions Recall that to multiply fractions, we multiply the numerators and multiply the denominators. The same is true for multiplying rational epressions. Multiplication of Rational Epressions Let p, q, r, and s represent polynomials, such that q 0 and s 0. Then p q r pr s qs For eample: Multiply the fractions 0 7 Factor. Multiply the rational epressions z 0z y 7 y Sometimes it is possible to simplify a ratio of common factors to before multiplying. To do so, we must first factor the numerators and denominators of each fraction The same process is also used to multiply rational epressions. Multiplying Rational Epressions. Factor the numerators and denominators of all rational epressions.. Simplify the ratios of common factors to.. Multiply the remaining factors in the numerator, and multiply the remaining factors in the denominator.
15 Section 6. Multiplication and Division of Rational Epressions 47 Eample Multiplying Rational Epressions Multiply. a b a. b. 0 a b 4w 0p w 0p w 7wp p w 9p Solution: a. a b 0 a b a b a ba b Factor numerator and denominator. b. a b a b 4w 0p w 0p w 7wp p w 9p 4w p w p w p w pw p a ba b w pw p w p w pw p w p Simplify. Factor numerator and denominator. Factor further. Avoiding Mistakes: If all factors in the numerator simplify to, do not forget to write the factor of in the numerator. w p w pw p w pw p w p w p w p Simplify common factors. Notice that the epression is left in factored form to show that it has been simplified to lowest terms. Skill Practice Multiply. y 6. y y. 6y y 4 p 8p 6 p 6 0p 0 p 7p. Division of Rational Epressions Recall that to divide fractions, multiply the first fraction by the reciprocal of the second fraction. Divide: Multiply by the reciprocal of the second fraction The same process is used for dividing rational epressions. Skill Practice Answers y p 4.. y p
16 48 Chapter 6 Rational Epressions and Rational Equations Division of Rational Epressions Let p, q, r, and s represent polynomials, such that q 0, r 0, s 0. Then p q r s p q s ps r qr Eample Dividing Rational Epressions 8t 7 Divide. a. b. 9 4t 4t 6t 9 t t Solution: a. 8t 7 9 4t 4t 6t 9 t t 8t 7 9 4t t t 4t 6t 9 t 4t 6t 9 t t t 4t 6t 9 t t t t or t t t 4t 6t 9 t t 4t 6t 9 c 6d 0 d Multiply the first fraction by the reciprocal of the second. Factor numerator and denominator. Notice 8t 7 is a sum of cubes. Furthermore, 4t 6t 9 does not factor over the real numbers. Simplify to lowest terms. TIP: In Eample (a), the factors (t ) and ( t) are opposites and form a ratio of. t t The factors (t ) and ( t) are equal and form a ratio of. t t c b. 6d 0 This fraction bar denotes division. d c 0 6d d c d 6d 0 Multiply the first fraction by the reciprocal of the second.
17 Section 6. Multiplication and Division of Rational Epressions 49 c 6d d d cd Skill Practice Divide the rational epressions rs t r t Skill Practice Answers s. 4. Section 6. Boost your GRADE at mathzone.com! Study Skills Eercise Practice Eercises Practice Problems SelfTests NetTutor eprofessors Videos. Write an eample of how to multiply two fractions, divide two fractions, add two fractions, and subtract two fractions. For eample: and Then as you learn about rational epressions, , 4 8, 9, 7 compare the operations on rational epressions to those on fractions. This is a great place to use cards again. Write an eample of an operation with fractions on one side and the same operation with rational epressions on the other side. Review Eercises For Eercises 9, simplify the rational epression to lowest terms. w t t 6 y w w 6 t 7t 6 y 6y yz yz p p p a b c a bc Concept : Multiplication of Rational Epressions For Eercises 0, multiply the rational epressions. 8w 6 z w 4 z 8 7r z 6z. 8rs 4.. 7s 9r s 8z 9z 6 p q 4 6p pq 0q y 4 y
18 40 Chapter 6 Rational Epressions and Rational Equations y 8y y 0w w w w 4 6y 6 y w 6 y y y 0y y yy 4 y Concept : Division of Rational Epressions For Eercises, divide the rational epressions.. 6 y y. 4. t t t t. a a 0 a 6a 40a a 00 y y 8. y y 9. 4 y y y y y.. y 7 y y y 9 r 4r s r rs 6p 7 p 6p 49 b 6b 9 b b 6 b 9 4 6s st t 6s st t s 7st 0t 6s st t a a a a a ab b y y 6 y a a a b b a ab ab b Mied Eercises For Eercises 4 4, perform the indicated operations. 8a 4 b y 4 y a7 b c 9c 7 6y 6 4y 7 y 4 y 0 y y y y y 8 y a b a b a b a b a ab b a b 4y y 4. y 4. y y 8 7y y 8 y 8y y 6a ab b 0a ab a 4a b a ab b m n m n m mn n m mn n 6y 9y y 6y 9y 4y y y m m m m m m n4 m n
19 Section 6. Addition and Subtraction of Rational Epressions 4 Epanding Your Skills For Eercises 46 49, write an epression for the area of the figure and simplify b in. a 8 hk cm 4a b in k h cm 4 m ft 4 6 m ft Addition and Subtraction of Rational Epressions. Addition and Subtraction of Rational Epressions with Like Denominators To add or subtract rational epressions, the epressions must have the same denominator. As with fractions, we add or subtract rational epressions with the same denominator by combining the terms in the numerator and then writing the result over the common denominator. Then, if possible, we simplify the epression to lowest terms. Addition and Subtraction of Rational Epressions Let p, q, and r represent polynomials where q 0. Then p p.. q r p r q r p r q q q q Section 6. Concepts. Addition and Subtraction of Rational Epressions with Like Denominators. Identifying the Least Common Denominator. Writing Equivalent Rational Epressions 4. Addition and Subtraction of Rational Epressions with Unlike Denominators Eample Adding and Subtracting Rational Epressions with Like Denominators Add or subtract as indicated. a. b. c
20 4 Chapter 6 Rational Epressions and Rational Equations Solution: a. Add the terms in the numerator Simplify the fraction. b. Add the terms in the numerator. The answer is already in lowest terms. c Combine the terms in the numerator. Use parentheses to group the terms in the numerator that follow the subtraction sign. This will help you remember to apply the distributive property. Apply the distributive property. Factor the numerator and denominator. 4 4 Simplify the rational epression. Skill Practice Add or subtract as indicated. b b 7... b b 4t t 7 t t 7. Identifying the Least Common Denominator If two rational epressions have different denominators, each epression must be rewritten with a common denominator before adding or subtracting the epressions. The least common denominator (LCD) of two or more rational epressions is defined as the least common multiple of the denominators. 0 For eample, consider the fractions and 8. By inspection, you can probably see that the least common denominator is 40. To understand why, find the prime factorization of both denominators. 0 and 8 Skill Practice Answers t.. b. t 7
21 Section 6. Addition and Subtraction of Rational Epressions 4 A common multiple of 0 and 8 must be a multiple of, a multiple of, and a multiple of. However, any number that is a multiple of 8 is automatically a multiple of 4. Therefore, it is sufficient to construct the least common denominator as the product of unique prime factors, where each factor is raised to its highest power. The LCD of and is 40. Steps to Find the LCD of Two or More Rational Epressions. Factor all denominators completely.. The LCD is the product of unique prime factors from the denominators, where each factor is raised to the highest power to which it appears in any denominator. Eample Finding the LCD of Rational Epressions Find the LCD of the following rational epressions. a. b. y ; ; 8 ; 7 0 6y z c. 4 d. ; 0 ; ; Solution: a. ; 8 ; 7 0 b. c. ; 7 ; LCD 80 y ; 6y z y ; 4 y z LCD 4 y z 6 y z 0 ; ; ; ; 6 4 LCD 4 Factor the denominators completely. The LCD is the product of the factors,, and. Each factor is raised to its highest power. Factor the denominators completely. The LCD is the product of the factors,, y, and z. Each factor is raised to its highest power. Factor the denominators completely. The LCD is the product of the factors,,, and 4. Each factor is raised to its highest power.
22 44 Chapter 6 Rational Epressions and Rational Equations d. 4 The denominators are already ; factored. Notice that and are opposite factors. If is factored from either epression, the binomial factors will be the same. 4 ; Factor out. same binomial factors LCD Factor out. same binomial factors The LCD can be taken as either or. Skill Practice Find the LCD of the rational epressions ; ; ; 6 ; ; LCD 9a b ; 8a 4 b 6 z 7 ; 7 z. Writing Equivalent Rational Epressions Rational epressions can be added if they have common denominators. Once the LCD has been determined, each rational epression must be converted to an equivalent rational epression with the indicated denominator. Using the identity property of multiplication, we know that for q 0 and r 0, p q p q p q r pr r qr This principle is used to convert a rational epression to an equivalent epression with a different denominator. For eample, can be converted to an equivalent epression with a denominator of as follows: In this eample, we multiplied by a convenient form of. The ratio 6 was chosen so that the product produced a new denominator of. Notice that multiplying by 6 is equivalent to multiplying the numerator and denominator of the 6 original epression by 6. 6 Eample Creating Equivalent Rational Epressions Skill Practice Answers a 4 b 6. ( )( )( ) 7. z 7 or 7 z Convert each epression to an equivalent rational epression with the indicated denominator. 7 w a. b. p w 0p 6 w w 0
23 Section 6. Addition and Subtraction of Rational Epressions 4 Solution: a. 7 p 0p 6 p 4p 4 7 p 7 4p4 p 4p 4 0p 6 can be written as p 4p 4. Multiply the numerator and denominator by the missing factor, 4p 4. 8p4 0p 6 b. w w w w 0 w w w w w w w w w w w w Factor the denominator. Multiply the numerator and denominator by the missing factor w. Skill Practice Fill in the blank to make an equivalent fraction with the given denominator. b y 6 b y b 9 TIP: In the final answer in Eample (b) we multiplied the polynomials in the numerator but left the denominator in factored form. This convention is followed because when we add and subtract rational epressions, the terms in the numerators must be combined. 4. Addition and Subtraction of Rational Epressions with Unlike Denominators To add or subtract rational epressions with unlike denominators, we must convert each epression to an equivalent epression with the same denominator. For eample, consider adding the epressions. The LCD is. For each epression, identify the factors from the LCD that are missing in the denominator. Then multiply the numerator and denominator of the epression by the missing factor(s): The rational epressions now have the same denominator and can be added. Skill Practice Answers 8. y 9. b b
24 46 Chapter 6 Rational Epressions and Rational Equations Combine terms in the numerator. Clear parentheses and simplify. Steps to Add or Subtract Rational Epressions. Factor the denominator of each rational epression.. Identify the LCD.. Rewrite each rational epression as an equivalent epression with the LCD as its denominator. 4. Add or subtract the numerators, and write the result over the common denominator.. Simplify if possible. Eample 4 Adding and Subtracting Rational Epressions with Unlike Denominators Add or subtract as indicated. t a. b. c. t 4t 7b 4 b t Solution: a. 7b 4 b Step : The denominators are already factored. Step : The LCD is 7b. 7b b b 4 b 7 7 b 8 7b 7b b 8 7b Step : Write each epression with the LCD. Step 4: Add the numerators, and write the result over the LCD. Step : Simplify. 8 b. t t 4t t t t 6t t 6 t t 6t t t 6 t Step : Factor the denominators. Step : The LCD is t 6t. Step : Write each epression with the LCD.
25 Section 6. Addition and Subtraction of Rational Epressions 47 c. t t t 6t 6t 4 t 0 t 6t t 6 t 6t t 6 t 6t t Step 4: Add the numerators and write the result over the LCD. Step : Simplify. Combine like terms. Simplify. Step : Factor the denominators. Step : The LCD is. Step : Write each epression with the LCD. Step 4: Add the numerators, and write the result over the LCD. Step : Simplify. Combine like terms. Factor the numerator. 4 4 Simplify to lowest terms. Skill Practice Add or subtract as indicated y y a a 4 a 9 a Skill Practice Answers 0. y. y. a a
26 48 Chapter 6 Rational Epressions and Rational Equations Eample Add or subtract as indicated. 6 a. b. y y w 4 w y Adding and Subtracting Rational Epressions with Unlike Denominators Solution: 6 a. Step : The denominators are already w 4 w factored. 6 w 4 w 6 w 4 w 6 4 w Step : The denominators are opposites and differ by a factor of. The LCD can either be taken as w or w. We will use an LCD of w. Step : Write each epression with the LCD. Note that w w. Step 4: Add the numerators, and write the result over the LCD. w Step : Simplify. b. Step : The denominators are already y y factored. y y y y Step : The denominators are opposites and differ by a factor of. The LCD can be taken as either y or y. We will use an LCD of y. Step : Write each epression with the LCD. Note that y y y. y y y y y y y y y Step 4: Combine the numerators, and write the result over the LCD. Step : Factor and simplify to lowest terms. Skill Practice Answers 8 8. or 4. y y Skill Practice. 4. y y Add or subtract as indicated. a a a
27 Section 6. Addition and Subtraction of Rational Epressions 49 Section 6. Boost your GRADE at mathzone.com! Study Skills Eercise Practice Eercises Practice Problems SelfTests NetTutor eprofessors Videos. Define the key terms. a. Least common denominator (LCD) b. Equivalent rational epressions Review Eercises For Eercises 6, perform the indicated operation. 9b 9 b y y 4b 8 b 8a 4a 7b b a. 6. 0a a a 0a z 4 z z z 4 z Concept : Addition and Subtraction of Rational Epressions with Like Denominators For Eercises 7 6, add or subtract as indicated and simplify if possible Concept : Identifying the Least Common Denominator For Eercises 7 8, find the least common denominator (LCD). y ; 0 a ; y ; 7 7y ; y.. 6m 4; mn 7 cd ; 9 8c ; ;
28 40 Chapter 6 Rational Epressions and Rational Equations ; ; ; 8 7a a 4 ; a a 6 Concept : Writing Equivalent Rational Epressions For Eercises 9 4, fill in the blank to make an equivalent fraction with the given denominator y y 4 y.. y. 4. y 6 y y 6 8 t t 8 t 6t 6 Concept 4: Addition and Subtraction of Rational Epressions with Unlike Denominators For Eercises 66, add or subtract as indicated a b p p 0ab s t 40. s t 4a a bb a a 4 b 6y y w 46. y y w 6b a a b 4 b a 6 s 8 6s s 8 s z z z 9 z z 4y n n n n ww w h h t 0t t t 7 w 6 y y w 4 6 c c 7 7 c c 7 4 k k 7 k 6k 7 4k k
29 Section 6.4 Comple Fractions 4 t y y t t t t t y w w w w w w w y y 6 8 p p p Epanding Your Skills For Eercises 67 70, write an epression that represents the perimeter of the figure and simplify cm cm yd yd 6 cm 4 yd m ft m ft Comple Fractions. Simplifying Comple Fractions by Method I A comple fraction is a fraction whose numerator or denominator contains one or more fractions. For eample: y 0 y and 4 6 are comple fractions. Two methods will be presented to simplify comple fractions. The first method (Method I) follows the order of operations to simplify the numerator and denominator separately before dividing. The process is summarized as follows. Section 6.4 Concepts. Simplifying Comple Fractions by Method I. Simplifying Comple Fractions by Method II. Using Comple Fractions in Applications
30 4 Chapter 6 Rational Epressions and Rational Equations Steps to Simplify a Comple Fraction Method I. Add or subtract epressions in the numerator to form a single fraction. Add or subtract epressions in the denominator to form a single fraction.. Divide the rational epressions from step by multiplying the numerator of the comple fraction by the reciprocal of the denominator of the comple fraction.. Simplify to lowest terms, if possible. Eample Simplifying a Comple Fraction by Method I Simplify the epression. Solution: y 0 y y y 0 y y y y y y y 0 y Step : The numerator and denominator of the comple fraction are already single fractions. Step : Multiply the numerator of the comple fraction by the reciprocal of the denominator. Factor the numerators and denominators. Step : Simplify. y Skill Practice. 8a b 6a b Simplify the epression. Sometimes it is necessary to simplify the numerator and denominator of a comple fraction before the division is performed. This is illustrated in Eample. Skill Practice Answers a. b
31 Section 6.4 Comple Fractions 4 Eample Simplifying a Comple Fraction by Method I Simplify the epression. 4 6 Solution: Step : Combine fractions in numerator and denominator separately. The LCD in the numerator is 6. The LCD in the denominator is. Form single fractions in the numerator and denominator. Step : Multiply by the reciprocal of, which is Step : Simplify.. Skill Practice. 4 6 Simplify the epression.. Simplifying Comple Fractions by Method II We will now use a second method to simplify comple fractions Method II. Recall that multiplying the numerator and denominator of a rational epression by the same quantity does not change the value of the epression. This is the basis for Method II. Skill Practice Answers. 7
32 44 Chapter 6 Rational Epressions and Rational Equations Steps to Simplify a Comple Fraction Method II. Multiply the numerator and denominator of the comple fraction by the LCD of all individual fractions within the epression.. Apply the distributive property, and simplify the numerator and denominator.. Simplify to lowest terms, if possible. Eample Simplifying Comple Fractions by Method II Simplify by using Method II. 4 6 Solution: 4 6 The LCD of all individual terms is. a4 6 b a b Step : Multiply the numerator and denominator of the comple fraction by the LCD, which is. 4 a 6 b a b 4 6 Step : Apply the distributive property. Step : Simplify numerator and denominator. Factor and simplify to lowest terms. Skill Practice Simplify the epression.. y y y Skill Practice Answers. y
33 Section 6.4 Comple Fractions 4 Eample 4 Simplifying Comple Fractions by Method II Simplify by using Method II. Solution: Rewrite the epression with positive eponents. The LCD of all individual terms is. a b a b Step : Multiply the numerator and denominator of the comple fraction by the LCD. a b a b a b a b Step : Apply the distributive property. Step : Factor and simplify to lowest terms. Skill Practice 4. c b b c b c Simplify the epression. Eample Simplifying Comple Fractions by Method II Simplify the epression by Method II. w w 9 w 9 Skill Practice Answers 4. b c
34 46 Chapter 6 Rational Epressions and Rational Equations Solution: w w 9 w 9 The LCD of w w w w 9 w w w w w w 9 w w Factor all denominators to find the LCD., w, w, and 9 is w w. w w w w a w w b 9 w w c w w d w w a w b w w a w b 9 w w w w c w w d Step : Simplify. Step : Multiply numerator and denominator of the comple fraction by w w. Apply the distributive property. Step : Distributive property. 6 w Skill Practice. Simplify the epression.. Using Comple Fractions in Applications Eample 6 Finding the Slope of a Line Find the slope of the line that passes through the given points. Skill Practice Answers. a, 6 b and a 9 8, 4 b
35 Section 6.4 Comple Fractions 47 Solution: Let, y a, 6 b and, y a 9 8, 4 b. Label the points. Slope y y Apply the slope formula. Using Method II to simplify the comple fractions, we have 4 a 4 6 b m 4 a 9 8 b Multiply numerator and denominator by the LCD, a b 4 4 a 4 6 b 4 a 9 b 4 a 8 b Apply the distributive property The slope is Skill Practice. Simplify. 6. Find the slope of the line that contains the points a and a 7, 4 b 0, b. Skill Practice Answers 6. m 6 Section 6.4 Boost your GRADE at mathzone.com! Study Skills Eercise. Define the key term comple fraction. Practice Eercises Practice Problems SelfTests NetTutor eprofessors Videos Review Eercises For Eercises 4, simplify to lowest terms. y a b c.. 4. f t 6t 7t 0 y a 4 bc t 0
36 48 Chapter 6 Rational Epressions and Rational Equations For Eercises 8, perform the indicated operations. y a y y y y 4 a 7 6b 4b b b 6 Concept : Simplifying Comple Fractions by Method I For Eercises 9 6, simplify the comple fractions by using Method I. w 6 9y 4rs wr 8 y s y y Concept : Simplifying Comple Fractions by Method II For Eercises 7 44, simplify the comple fractions by using Method II. 7y 6 y y q b b a p q a a q q b b a 9 p a a t d c t c d 6w w 4 y 4 6 y w y 4 4 y y w 4 a t p a a t p a h t 4 y 4 y t h t y 7 y t 4 a b a b a a 4 0 y y 4 y y 6 z 0 z 4 9 b 4 b 4 b b h h m n m 4n
37 Section 6.4 Comple Fractions 49 Concept : Using Comple Fractions in Applications 4. The slope formula is used to find the slope of the line passing through the points, y and, y. Write the slope formula from memory. For Eercises 46 49, find the slope of the line that passes through the given points. 46. a 47. a 48. a 49. 8, 9 b, a 7, b,,, b, a 4, b 0 6, b a 4, b, a 8, 6 b Epanding Your Skills 0. Show that by writing the epression on the left without negative eponents and simplifying.. Show that y y by writing the epression on the left without negative eponents and y simplifying.. Simplify.. Simplify. a a b b Chapter 6 Problem Recognition Eercises Operations on Rational Epressions For Eercises 0, perform the indicated operation and simplify. 7.. a y y 4 b a a ab a 4a ab 4b a 9a 0 4 y y y y 8w w 6w 4w w 4w y y y y 6 8 a b w w w y 6 y y 0 y y y 7 y a a 4 t 9 t t t 6. t t a a 6 y y
38 440 Chapter 6 Rational Epressions and Rational Equations y y y 4 y y 8. y z 4 6y7 0z 0. a 0a 00 a 0 a 0 Section 6. Rational Equations Concepts. Solving Rational Equations. Formulas Involving Rational Equations. Solving Rational Equations Thus far we have studied two types of equations in one variable: linear equations and quadratic equations. In this section, we will study another type of equation called a rational equation. Definition of a Rational Equation An equation with one or more rational epressions is called a rational equation. The following equations are rational equations: 4 6w w 6 w To understand the process of solving a rational equation, first review the procedure of clearing fractions from Section.4. Eample Solve the equation. Solving a Rational Equation 4 Solution: 4 The LCD of all terms in the equation is. a b a 4 b Multiply both sides by to clear fractions Apply the distributive property. Solve the resulting equation. 4 Check: 4 a4 b 4 a4 b
39 Section 6. Rational Equations 44 Skill Practice. 0 Solve the equation. The same process of clearing fractions is used to solve rational equations when variables are present in the denominator. Eample Solve the equation. Solving a Rational Equation Solution: The LCD of all terms in the equation is. a b a b Multiply by to clear fractions. 9 0 Apply the distributive property. Solve the resulting equation. Check: 9 0 Skill Practice. Solve. y 4 Eample Solving a Rational Equation Solve the equation. 6w w 6 w Skill Practice Answers. 7. y 9 7
40 44 Chapter 6 Rational Epressions and Rational Equations Solution: 6w w 6 w w w a 6w w b w a 6 w b w w a 6w w b w a 6 w b w 6w 6 w w The LCD of all terms in the equation is w. Multiply by w on both sides to clear fractions. Apply the distributive property. Solve the resulting equation. Check: 6w w 6 w 6 6 Because the value w makes the denominator zero in one (or more) of the rational epressions within the equation, the equation is undefined for w. No other potential solutions eist, so the equation 6w has no w 6 w solution. Skill Practice. The denominator is 0 for the value of w. Solve the equation. 4 b b 8 b 4 Eamples show that the steps for solving a rational equation mirror the process of clearing fractions from Section.4. However, there is one significant difference. The solutions of a rational equation must be defined in each rational 6w epression in the equation. When w is substituted into the epression w 6 or, the denominator is zero and the epression is undefined. Hence w w cannot be a solution to the equation 6w w 6 w The steps for solving a rational equation are summarized as follows. Skill Practice Answers. No solution (b does not check.)
41 Section 6. Rational Equations 44 Steps for Solving a Rational Equation. Factor the denominators of all rational epressions. Identify any values of the variable for which any epression is undefined.. Identify the LCD of all terms in the equation.. Multiply both sides of the equation by the LCD. 4. Solve the resulting equation.. Check the potential solutions in the original equation. Note that any value from step for which the equation is undefined cannot be a solution to the equation. Eample 4 Solving Rational Equations Solve the equations. a. 8 b. Solution: a. 8 The LCD of all terms in the equation is. Epressions will be undefined for 0. Check: 7 Check: 4 a b a 8 b or p 9 p p Multiply both sides by Apply the distributive property. to clear fractions. The resulting equation is quadratic. Set the equation equal to zero and factor Both solutions check. 6 6
42 444 Chapter 6 Rational Epressions and Rational Equations b. 6 p 9 p p 6 p p p p 6 pp p p 6 p 6p p 9 6 p 6p p 9 6 p 6p 9 0 p 6p 7 0 p 9p p 9 or p The LCD is p p. Epressions will be undefined for p = and p =. Multiply both sides by the LCD to clear fractions. 6 p p c d p p a p b p p p p p 6 p p c d p p a p b p p p p p Solve the resulting equation. The equation is quadratic. Set the equation equal to zero and factor. p is not in the domain of the original epressions. Therefore, it cannot be a solution. This can be verified when we check. Check: p 9 6 p 9 p p Check: p 6 p 9 p p Denominator is zero. The solution is p 9. Skill Practice Solve the equations. y y 6 Skill Practice Answers 4. y 4; y. 9 does not check.)
43 Section 6. Rational Equations 44. Formulas Involving Rational Equations Eample Solving Literal Equations Involving Rational Epressions Solve for the indicated variable. a. P A for r b. V mv rt m M for m Solution: a. P A rt for r A rtp rta rt b A rtp rta rt b P Prt A Prt A P Prt Pt A P Pt r A P Pt b. V mv m M for m mv Vm M a bm M m M Vm M mv Vm VM mv Vm mv VM mv v VM mv v VM V v V v m VM V v Multiply both sides by the LCD rt. Clear fractions. Apply the distributive property to clear parentheses. Isolate the rterm on one side. Divide both sides by Pt. Multiply by the LCD and clear fractions. Use the distributive property to clear parentheses. Collect all m terms on one side. Factor out m. Divide by V v. Avoiding Mistakes: Variables in algebra are casesensitive. Therefore, V and v are different variables, and M and m are different variables. TIP: The factor of that appears in the numerator may be written in the denominator or out in front of the epression. The following epressions are equivalent: m VM V v or m VM V v VM or m VM V v v V
44 446 Chapter 6 Rational Epressions and Rational Equations Skill Practice Answers 6. q rs or q rs r s s r b yd yd b 7. or y a a y Skill Practice 6. Solve the equation for q. 7. Solve the equation for. q r s y a b d Section 6. Boost your GRADE at mathzone.com! Study Skills Eercise. Define the key term rational equation. Practice Eercises Practice Problems SelfTests NetTutor eprofessors Videos Review Eercises For Eercises 8, perform the indicated operations y y y t 7t. t t m 9 m m m m y y 8. 0 Concept : Solving Rational Equations For Eercises 9 40, solve the rational equations. 6 y y w 0 4p w a 4a 0 y y y w 8 4w p t t 0. a 4a 6. w w 7. 8t t 8. 6z z 0
45 Section 6. Rational Equations t 8 t t y 0 y 4 y y k k 6k 4 k h h h 4 4 h Concept : Formulas Involving Rational Equations For Eercises 4 8, solve the formula for the indicated variable. 4. K ma for m 4. K ma F F for a K IR for R 4. I E E R r for R 46. K IR E I E R r for E for r 47. V at b h A for B ph r for h B b t for t 0. T mf y for m. z for. w n m g y wn P for w. a b A for h 4. rt A for P. R h P R R for R 6. b a for b 7. for t 8. a v v v s s ab f t t t t for v Mied Review 0 9. a. Simplify. 60. a. Simplify. w w w 0 b. Solve. b. Solve. w w w c. What is the difference in the type of problem c. What is the difference in the type of given in parts (a) and (b)? problem given in parts (a) and (b)?
46 448 Chapter 6 Rational Epressions and Rational Equations For Eercises 6 70, perform the indicated operation and simplify, or solve the equation for the variable a 4a a c 6 4 c 8c y y y y w 7 w 4p p p p b b b b 0 t 0 4 t 4 y y 4 Epanding Your Skills 7. If is added to the reciprocal of a number, the result is. Find the number. 7. If is added to the reciprocal of a number, the result is. Find the number. 7. If 7 is decreased by the reciprocal of a number, the result is. Find the number. 74. If a number is added to its reciprocal, the result is 6. Find the number Section 6.6 Concepts. Solving Proportions. Applications of Proportions. Similar Triangles 4. Applications of Rational Equations Applications of Rational Equations and Proportions. Solving Proportions A proportion is a rational equation that equates two ratios. Definition of Ratio and Proportion a. The ratio of a to b is b 0 and can also be epressed as b a : b or a b.. An equation that equates two ratios or rates is called a proportion. Therefore, if b 0 and d 0, then a is a proportion. b c d The process for solving rational equations can be used to solve proportions.
47 Section 6.6 Applications of Rational Equations and Proportions 449 Eample Solving a Proportion Solve the proportion. Solution: 9 9 y 9y a b 9y a9 9 y b 9y a b 9y a9 9 y b y 80 y 80 y y The LCD is 9y. Note that y 0. Multiply both sides by the LCD. Clear fractions. Solve the resulting equation. The solution checks in the original equation. Skill Practice. 8 Solve the proportion. TIP: For any proportion a b c d b 0, d 0 the cross products of terms are equal. Hence, ad bc. Finding the cross product is a quick way to clear fractions in a proportion.* Consider Eample : 9 9 y y 99 y 80 y 6 Equate the cross products. *It is important to realize that this method is only valid for proportions.. Applications of Proportions Eample Solving a Proportion In the U.S. Senate in 006 the ratio of Democrats to Republicans was 4 to. If there were 44 Democrats, how many Republicans were there? Skill Practice Answers. or 7.
48 40 Chapter 6 Rational Epressions and Rational Equations Solution: One method of solving this problem is to set up a proportion. Write two equivalent ratios depicting the number of Democrats to the number of Republicans. Let represent the unknown number of Republicans in the Senate. Given ratio 4 44 Number of Democrats Number of Republicans a 4 b a44 b Multiply both sides by the LCD. Clear fractions. There were Republicans in the U.S. Senate in 006. Skill Practice The ratio of cats to dogs at an animal rescue facility is 8 to. How many dogs are in the facility if there are 400 cats? Eample Solving a Proportion The ratio of male to female police officers in a certain town is :. If the total number of officers is, how many are men and how many are women? Solution: Let represent the number of male police officers. Then represents the number of female police officers. Male Female a b a b Number of males Number of females The number of male police officers is 88. The number of female officers is Multiply both sides by. The resulting equation is linear. Skill Practice Answers. 0 dogs. passed and 9 failed. Skill Practice. Professor Wolfe has a ratio of passing students to failing students of to 4. One semester he had a total of 07 students. How many students passed and how many failed?
49 Section 6.6 Applications of Rational Equations and Proportions 4. Similar Triangles Proportions are used in geometry with similar triangles. Two triangles are said to be similar if their corresponding angles are equal. In such a case, the lengths of the corresponding sides are proportional. The triangles in Figure 6 are similar. Therefore, the following ratios are equivalent. a b y c z c a z b Figure 6 y Eample 4 Using Similar Triangles in an Application The shadow cast by a yardstick is ft long. The shadow cast by a tree is ft long. Find the height of the tree. Solution: Let represent the height of the tree. We will assume that the measurements were taken at the same time of day. Therefore, the angle of the sun is the same on both objects, and we can set up similar triangles (Figure 66). yd ft ft Figure 66 ft Height of yardstick Length of yardstick s shadow The tree is 6. ft high. ft ft ft a b a b 6. Height of tree Length of tree s shadow Write an equation. Multiply by the LCD. Solve the equation. Interpret the results.
50 4 Chapter 6 Rational Epressions and Rational Equations Skill Practice 4. Solve for, given that the triangles are similar. 9 m m 4 m 4. Applications of Rational Equations Eample Solving a Business Application of Rational Epressions To produce mountain bikes, the manufacturer has a fied cost of $6,000 plus a variable cost of $40 per bike. The cost in dollars to produce bikes is given by C 6, The average cost per bike C C is defined as C or Find the number of bikes the manufacturer must produce so that the average cost per bike is $80. Solution: Given C 80 C 6, , , Replace C by 80 and solve for. 6, Multiply by the LCD. 80 6, The resulting equation is linear. 40 6,000 Solve for. 6, The manufacturer must produce 400 bikes for the average cost to be $80. Skill Practice Answers 4. m. 960 pairs of skis Skill Practice. To manufacture ski equipment, a company has a fied cost of $48,000 plus a variable cost of $00 per pair of skis. The cost to produce pairs of skis is C() 48, The average cost per pair of skis is 48, C Find the number of sets of skis the manufacturer must produce so that the average cost per set is $0.
51 Section 6.6 Applications of Rational Equations and Proportions 4 Eample 6 Solving an Application Involving Distance, Rate, and Time An athlete s average speed on her bike is 4 mph faster than her average speed running. She can bike. mi in the same time that it takes her to run 0. mi. Find her speed running and her speed biking. Solution: Because the speed biking is given in terms of the speed running, let represent the running speed. Let represent the speed running. Then 4 represents the speed biking. Organize the given information in a chart. Distance Rate Time Running 0. Biking Because d rt, then t d r The time required to run 0. mi is the same as the time required to bike. mi, so we can equate the two epressions for time: 4a 0.. b 4a 4 b The LCD is 4. Multiply by 4 to clear fractions The resulting equation is linear Solve for The athlete s speed running is 7 mph. The speed biking is 4 or 7 4 mph. Skill Practice Devon can crosscountry ski km/hr faster than his sister, Shanelle. Devon skis 4 km in the same amount of time that Shanelle skis 0 km. Find their speeds. Skill Practice Answers 6. Shanelle skis 0 km/hr and Devon skis km/hr.
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