4-4 Adding and Subtracting Rational Expressions PearsonRealize.com I CAN… find the sum or difference of rational expressions. VOCABULARY • compound fraction ESSENTIAL QUESTION Activity CRITIQUE & EXPLAIN Teo and Shannon find the following exercise in their homework: __ 1 + __ 1 + __ 1 2 Compare addition of numerical and algebraic fractions: +3 _ _ 15 + _ 35 = 1___ = 45 5 In the same way, +5 x ____ + ____ x +5 4 = ____ xx + . x+4 4 3 9 A. Teo claims that a common denominator of the sum is 2 + 3 + 9 = 14. Shannon claims that it is 2 ∙ 3 ∙ 9 = 54. Is either student correct? Explain why or why not. 1+1+1 2 3 9 B. Find the sum, explaining the method you use. C. Construct Arguments Timothy states that the quickest way to find the sum of any two fractions with unlike denominators is to multiply their denominators to find a common denominator, and then rewrite each fraction with that denominator. Do you agree? How do you rewrite rational expressions to find sums and differences? Add Rational Expressions With Like Denominators EXAMPLE 1 USE STRUCTURE Assess What is the sum? x + _____ A. _____ 5 x+4 x+4 x+5 = _____ When denominators are the same, add the numerators. x+4 So _____ x + _____ 5 = _____ x + 5 x+4 x+4 x+4 2x + 1 B. _______ + ________ 3x − 8 2 x + 3x x(x + 3) (2x + 1) + (3x − 8) x + 3x ________________ = 2 Add the numerators. (2x + 3x) + (1 − 8) x + 3x ________________ = 2 Use the Commutative and Associative Properties. − 7 = _______ 5x 2 Combine like terms. x + 3x +1 − 7 So _______ 2x + _______ 3x − 8 = _______ 5x 2 2 x + 3x x(x + 3) x + 3x Try It! 1. Find the sum. 10x − 5 + ______ a. _______ 8 − 4x 2x + 3 2x + 3 x − 5 + _______ b. _____ 3x − 21 x+5 x+5 LESSON 4-4 Adding and Subtracting Rational Expressions 217 Activity CONCEPTUAL UNDERSTANDING Assess Identify the Least Common Multiple of Polynomials EXAMPLE 2 How can you find the least common multiple (LCM) of polynomials? A. (x + 2) 2, x 2 + 5x + 6 B. x 3 − 9x, x 2 − 2x − 15, x 2 − 5x STUDY TIP Factor each polynomial. Factor each polynomial. Using the LCM of the denominators can mean less work simplifying later. (x + 2) 2 = (x + 2)(x + 2) x 3 − 9x = x(x 2 − 9) = x(x + 3)(x − 3) x 2 + 5x + 6 = (x + 2)(x + 3) x 2 − 2x − 15 = (x + 3)(x − 5) The LCM is the product of the factors. Duplicate factors are raised to the greatest power represented. x 2 − 5x = x(x − 5) LCM: x(x + 3)(x − 3)(x − 5) Each original polynomial is a factor of the LCM. LCM: (x + 2)(x + 2)(x + 3) or (x + 2) 2(x + 3) Try It! 2. Find the LCM for each set of expressions. a. x 3 + 9x 2 + 27x + 27, x 2 − 4x − 21 b. 10x 2 − 10y 2, 15x 2 − 30xy + 15y 2, x 2 + 3xy + 2y 2 Add Rational Expressions With Unlike Denominators EXAMPLE 3 What is the sum of_____ x2+ 3 and _________ 2 2 ? x − 1 USE STRUCTURE The LCM of 6 and 15 is 30, not 90. 1 _ 16 + _ 15 = ___ 2 1∙ 3 + ___ 3 1∙ 5 = ______ 1 ∙25∙ +3 ∙15∙ 2 x − 3x + 2 Follow a similar procedure to the one you use to add numerical fractions with unlike denominators. x+3 2 2 ______ x + 3 + __________ = ___________ + ___________ 2 − 1 x (x + 1)(x − 1) 2 − 3x + 2 x (x + 3)(x − 2) The LCM does not contain the common factor 3 twice. In the example problem, the common factor (x − 1) is not used twice. Factor each denominator. (x − 1)(x − 2) 2(x + 1) = _________________ + _________________ Use the LCM as (x + 1)(x − 1)(x − 2) (x + 1)(x − 1)(x − 2) the least common denominator (LCD). (x + 3)(x − 2) + 2(x + 1) (x + 1)(x − 1)(x − 2) Add the numerators. (x 2 + x − 6) + (2x + 2) (x + 1)(x − 1)(x − 2) Distribute. = ____________________ ___________________ = 2 x + 3x − 4 _________________ = Combine like terms. (x + 1)(x − 1)(x − 2) (x + 4)(x − 1) (x + 1)(x − 1)(x − 2) = _________________ (x + 4) (x + 1)(x − 2) Factor. (x − 1) (x − 1) = ___________ ∙ ______ Rewrite to identify unit factors. x+4 = ___________ for x ≠ −1, 1, and 2 Simplify and (x + 1)(x − 2) state the domain. x+3 x+4 2 _____ ________ __________ The sum of 2 and 2 is for x ≠ −1, 1, and 2. x −1 x −3x + 2 (x + 1)(x − 2) Try It! 3. Find the sum. x+6 a. ______ + __________ 2 2 2 x − 4 218 TOPIC 4 Rational Functions x − 5x + 6 2 2x + x − 11x − 12 _____________ b. ______ 4 2 3x + 4 6x + 5x − 4 Go Online | PearsonRealize.com Activity Assess Subtract Rational Expressions EXAMPLE 4 What is the difference between __________ 2 x + 1 and _________ 2 x + 1 ? x + 6x + 8 x − 6x − 16 x+1 x+1 x+1 ___________ − __________ 2 x + 1 = ___________ − ___________ 2 x − 6x − 16 (x − 8)(x + 2) x + 6x + 8 (x + 2)(x + 4) (x + 1)(x + 4) (x − 8)(x + 2)(x + 4) The LCD is (x − 8)(x + 2) (x + 4). (x − 8)(x + 1) (x − 8)(x + 2)(x + 4) _________________ = _________________ − (x 2 + 5x + 4) − (x 2 − 7x − 8) (x − 8)(x + 2)(x + 4) ________________________ = COMMON ERROR When subtracting polynomials, remember to distribute −1 when removing the parentheses. Check for common x 2 + 5x + 4 − x 2 + 7x + 8 _____________________ = factors in the numerator (x − 8)(x + 2)(x + 4) and denominator and 12x + 12 = _________________ simplify, if possible. (x − 8)(x + 2)(x + 4) 12(x + 1) (x − 8)(x + 2)(x + 4) _________________ = 12(x + 1) _______________ The difference between __________ 2 x + 1 and _________ 2 x + 1 is (x − 8)(x + 2)(x + 4) x − 6x − 16 x + 6x + 8 for x ≠ −4, −2, and 8. Try It! 4. Simplify. 3x − 5 − _____ b. _______ 2 2 1 + ___ a. ___ 1 − ___ 12 3x APPLICATION 6x x − 25 x x+5 Find a Rate EXAMPLE 5 Leah drives her car to the mechanic, then she takes the commuter rail train back to her neighborhood. The average speed for the 10-mile trip is 15 miles per hour faster on the train. Find an expression for Leah’s total travel time. If she drove 30 mph, how long did this take? Speed = r + 15 Speed = r USE APPROPRIATE TOOLS Use a table to organize information and help create an accurate model of the situation. Distance Rate Time Car 10 r Commuter Rail 10 r + 15 10 r 10 r + 15 Remember: distance = rate ∙ time, so time = ______ distance . rate Total time for the trip: 10(r + 15) ________ 10 = _________ ___ ______ 10 + 10r r + r + 15 r(r + 15) r(r + 15) Add the times for each part of the trip. 10r + 150 + 10r ______________ = r(r + 15) = _________ 20r + 150 r(r + 15) CONTINUED ON THE NEXT PAGE LESSON 4-4 Adding and Subtracting Rational Expressions 219 Activity Assess EXAMPLE 5 CONTINUED At a driving rate of 30 mph, you can find the total time. 20(30) + 150 30(30 + 15) ______ = 750 1,350 5 = __ 9 20r + 150 ___________ _________ = r(r + 15) Substitute 30 mph for the rate, and simplify. The expression for Leah’s total travel time is _______ 20r + 150 . The total time is __59 h, or r(r + 15) about 33 min. Try It! 5. On the way to work Juan carpools with a fellow co-worker, then takes the city bus back home in the evening. The average speed of the 20-mile trip is 5 miles per hour faster in the carpool. Write an expression that represents Juan’s total travel time. EXAMPLE 6 Simplify a Compound Fraction A compound fraction is in the form of a fraction and has one or more fractions in the numerator and/or the denominator. How can you write a simpler form of a compound fraction? __ x1 + _____ 2 x+ 1 ________ 1 __ y Method 1 F ind the Least Common Multiple (LCM) of the fractions in the numerator and denominator. Multiply the numerator and the denominator by the LCM. 2 ∙ [x(x + 1)y] 1 + _____ 2 [__ 1x + _____ __ x x + 1 x + 1] ________ ____________________ = __ __ 1y 1y ∙ [x(x + 1)y] Multiply the numerator and the denominator by the LCD. (x + 1)y + 2xy x(x + 1) = _____________ se the Distributive Property U to eliminate the fractions. xy + y + 2xy x(x + 1) = ___________ Simplify. (3x + 1)y x(x + 1) = _________ Factor. Method 2 E xpress the numerator and denominator as single fractions. Then multiply the numerator by the reciprocal of the denominator. 1 + _____ 2 __ x x + 1 ________ = __ 1y x + 1 _____ 1 ∙ _____ 2 x __ __ x x + 1 + x + 1 ∙ x _______________ __ 1y (x + 1) + 2x __________ x(x + 1) __________ = __ 1y USE STRUCTURE Simplify. y 1 3x + 1 __ = _______ ∙ Remember that the fraction bar separating the numerator and denominator represents division. x(x + 1) (3x + 1)y x(x + 1) Multiply the numerator by the reciprocal of the denominator. ________ = Using either method, x ≠ −1, 0 and y ≠ 0. __1x + ____ x +2 1 _______ __ 1y Simplify. 1 ____ x − 1 a. _________ x+1 4 ____ ____ 3 + x − 1 (3x + 1)y x(x + 1) is equal to _______ when Try It! 6. Simplify each compound fraction. 220 TOPIC 4 Rational Functions Multiply the numerator by the LCD. 1 __ 2 − x b. _____ x + __ 2x Go Online | PearsonRealize.com Concept Summary Assess CONCEPT SUMMARY Find Sums and Differences of Rational Expressions WORDS To add or subtract rational expressions with common denominators, add the numerators and keep the denominator the same. To add or subtract rational expressions with different denominators, rewrite each expression so that its denominator is the LCD, then add or subtract the numerators. NUMBERS 3 __ 1 + __ __ 3 + _____ 1 + = 4 __ 1 + __ 1 = ____ 1 + ____ 1 5 5 5 5 6 15 2∙3 3∙5 1∙5+1∙2 = __________ 2∙3∙5 ALGEBRA _____ x + _____ 5 = _____ x + 5 x+4 x+4 2 ______ x + 3 + __________ x+4 2 − 1 x Rewrite the rational expressions using the LCD. 2 − 3x + 2 x x+3 2 = ___________ + ___________ (x + 1)(x − 1) (x − 1)(x − 2) (x + 3)(x − 2) (x + 1)(x − 1)(x − 2) 2(x + 1) (x + 1)(x − 1)(x − 2) _________________ = + _________________ Do You UNDERSTAND? 1. Do You KNOW HOW? 6. Find the sum of ____ x +3 1 + ____ x 11 . +1 How do you rewrite rational expressions to find sums and differences? ESSENTIAL QUESTION Find the LCM of the polynomials. 2. Vocabulary In your own words, define compound fraction and provide an example of one. 3. Error Analysis A student added the rational expressions as follows: 7(7) x+7 5x + _____ 5x + _____ _____ 7 = _____ = _______ 5x + 49 x+7 x x+7 7.x 2 − y 2and x 2 − 2xy + y 2 8.5x 3yand 15x 2y 2 Find the sum or difference. y 4y 10x 9y + 2 __________ 3x − ____ 9. ____ 2 10. x+7 Describe and correct the error the student made. 4. Construct Arguments Explain why, when stating the domain of a sum or difference of rational expressions, not only should the simplified sum or difference be considered but the original expression should also be considered. 3y 2 − 2y − 8 + __________ 2 7 3y + y − 4 11. Find the perimeter of the quadrilateral in simplest form. x 2 3x 1 x 2x 5. Make Sense and Persevere In adding or subtracting rational expressions, why is the L in LCD significant? LESSON 4-4 Adding and Subtracting Rational Expressions 221 UNDERSTAND 13. Error Analysis Describe and correct the error a student made in adding the rational expressions. 1 + 2x 1 2x = + 3x + 3 (x + 2)(x + 1) 3(x + 1) 3 x(x + 2) = + 3(x + 2)(x + 1) 3(x + 2)(x + 1) = 3 + x2 + 2x 3(x + 2)(x + 1) = x2 + 2x + 3 3(x + 2)(x + 1) = (x + 3) (x + 1) • 3(x + 2) (x + 1) = x+3 3(x + 2) ✗ 14. Higher Order Thinking Find the slope of the line that passes through the points shown. Express in simplest form. 4 2 −4 −2 O −2 − 1 , −1 b a ( Find the sum. Additional Exercises Available Online y x+7 x+7 3y − 1 9y + 6 _______ _______ 18. y 2 + 4y + y(y + 4) Find the LCM for each group of expressions. SEE EXAMPLE 2 19. x 2 − 7x + 6, x 2 − 5x − 6 20. y 2 + 2y − 24, y 2 − 16, 2y Find the sum. SEE EXAMPLE 3 21. _______ 6x + _______ 4 x 2 − 8x 2x − 16 Find the difference. 3y 22. _______ + ___ 2 2 2y − y SEE EXAMPLE 4 y−1 3y + 15 2y y+3 5y + 25 24. _______ − _______ 25 On Saturday morning, Ahmed decided to take a bike ride from one end of the 15-mile bike trail to the other end of the bike trail and back. His average speed the first half of the ride was 2 mph faster than his speed on the second half. Find an expression for Ahmed’s total travel time. If his average speed for the first half of the ride was 12 mph, how long was Ahmed’s bike ride? SEE EXAMPLE 5 Speed x+2 ( 1a , 1b ( x 2 SEE EXAMPLE 1 17. _____ 4x + _____ 9 23. ______ 24x − _____ 4 x − 1 x − 1 (x + 3)(x + 1) = 3(x + 2)(x + 1) ( Tutorial PRACTICE 12. Generalize Explain how addition and subtraction of rational expressions is similar to and different from addition and subtraction of rational numbers. x2 + 3x + 2 Practice Scan for Multimedia PRACTICE & PROBLEM SOLVING Trail length 15 miles Speed x 4 −2 15. Reason For what values of x is the sum of x − 5y x + 7y _____ x + y and _____ x + y undefined? Explain. 16. Error Analysis A student says that the LCM of 3x 2 + 7x + 2and 9x + 3 is (3x 2 + 7x + 2) (9x + 3). Describe and correct the error the student made. 222 TOPIC 4 Rational Functions Rewrite as a rational expression. SEE EXAMPLE 6 1 + __ 1x 26. _____ x − __ 1x 3 + __ 7 __ y x 27. _____ 2 __ 1 − __ 1 + __ 1 __ a b _______ 28. 2 2 _______ a − b z − z −12 __________ 2 − 2z −15 z ___________ 29. __________ z 2 + 8z +12 2 ab y x 2 z −5z −14 Go Online | PearsonRealize.com Practice PRACTICE & PROBLEM SOLVING Mixed Review Available Online ASSESSMENT PRACTICE APPLY 30. Use Structure Aisha paddles a kayak 5 miles downstream at a rate 3 mph faster than the river’s current. She then travels 4 miles back upstream at a rate 1 mph slower than the river’s current. Hint: Let x represent the rate of the river current. 33. Which of the following compound fractions simplifies to ____ xx +– 31 ? Select all that apply. 2 2 x + 5x + 4 __________ x 2 + 2x −8 __________ Ⓐ 2 _________ x 2 − 4x + 3 x − 1 ______ 2 x − 4 Ⓑ __________ 2 __________ x + x − 7 2 + 5x + 6 x x − 3x + 2 2 2 x + 3x − 10 ___________ x 2 − 5x + 6 ___________ Ⓓ 2 _________ 2x − 25 x + 3x − 10 ___________ x 2 − 16 ___________ Ⓒ 2 5 __________ x −2 4x − Speed = x + 3 x − 1 water current = x mph a. Write and simplify an expression to represent the total time it takes Aisha to paddle the kayak 5 miles downstream and 4 miles upstream. b. If the rate of the river current, x, is 2 mph, how long was Aisha’s entire kayak trip? 31. Model With Mathematics Rectangles A and B are similar. An expression that represents the width of each rectangle is shown. Find the scale factor of rectangle A to rectangle B in simplest form. x2 − 25 x−4 Tutorial A B x+5 x2 − 16 32. Reason The Taylor family drives 180 miles (round trip) to a professional basketball game. On the way to the game, their average speed is approximately 8 mph faster than their speed on the return trip home. a. Let x represent their average speed on the way home. Write and simplify an expression to represent the total time it took them to drive to and from the game. b. If their average speed going to the game was 72 mph, how long did it take them to drive to the game and back? x − 4x − 5 34. SAT/ACT What is the difference between x−y 6 ? __9x and ____ 5x − y Ⓐ ______ 18 5x + y Ⓑ ______ 18 −x + 3y Ⓒ _______ 18 −x − 3y Ⓓ _______ 18 1 35. Performance Task The lens equation __1 = __ + ___ 1 f d i d o represents the relationship between f, the focal length of a camera lens, d i, the distance from the lens to the film, and d o, the distance from the lens to the object. Lens Object f do Film f di Part A Find the focal length of a camera lens if an object that is 12 cm from a camera lens is in focus on the film when the lens is 6 cm from the film. Part B Suppose the focal length of another camera lens is 3 inches, and the object to be photographed is 5 feet away. What distance (to the nearest tenth inch) should the lens be from the film? LESSON 4-4 Adding and Subtracting Rational Expressions 223
