11.4 Simplifying Rational Expressions. 669. 39. MULTI-STEP PROBLEM In this exercise you will look for a pattern. a. Copy and complete the table by evaluating the expression for each x-value. b. Describe the relationship between and x + 2. What does t
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All rights reserved. Simplifying Rational Expressions. Totally Rational. ACTIVITY 28 PRACTICE. Write your answers on notebook paper. Show your work. Lesson 28-1. 1. Allison correctly simplified the rational expression shown below by dividing. 35. 15.
Radical Expressions and Rational Exponents. Practice and Problem Solving: A/B. Write each expression in ... Simplify numerical expressions ... LESSON. 11-1.
May 4, 2014 - Have them rewrite the problem, multiplying the factors in each numerator and denominator. Then have them exchange problems with another pair, and find the quotient. Have each pair compare their answer to the answer determined by the stu
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OBJECTIVE. 1. Interactive lesson includes instant self-check, tutorials, and activities. Multiplying Rational Expressions. Lesson 12-4 Multiplying and Dividing Rational Expressions. 657. Need Help? Remember that the value of the expression in the den
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Radical Expressions and Rational Exponents. Reteach. = m n m n a a. Translate the expressions with rational exponents into radical expressions, then simplify.
May 4, 2014 - Class. Date. Explore Relating Multiplication Concepts. Use the facts you know about multiplying rational numbers to determine how to multiply rational ... 2 (x + 1) ? Reflect. 1. Discussion Multiplying rational expressions is similar to
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x x ax a x ax. x a. -. +. -. -. -. +. 4. Reduce. a b. b a. -. -. 5. Reduce. 2. 25. 5 x x. -. -. 6. Reduce. 3. 2. 3. 2. 2. 2. 24. 2. 8 x x x x x x. +. -. +. -. Multiplying and Dividing ...
Mar 21, 2014 - find this lesson in the hardcover student edition. Adding and. Subtracting Rational. Expressions. ENGAGE. Essential Question: How can you add and subtract rational expressions? Possible answer: Convert them to like denominators, then a
Additions and changes to the original content are the responsibility of the instructor. 164. Multiplying and Dividing Rational Expressions. Practice and Problem ...
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11.5 What you should learn GOAL 1 Multiply and divide rational expressions. GOAL 2 Use rational expressions as real-life models, as when comparing parts of the service industry to the total in Exs. 38–41.
Why you should learn it To model real-life situations, such as describing the average car sales per dealership in Example 6.
Multiplying and Dividing Rational Expressions GOAL 1
FINDING PRODUCTS AND QUOTIENTS
Because the variables in a rational expression represent real numbers, the rules for multiplying and dividing rational expressions are the same as the rules for multiplying and dividing numerical fractions.
M U LT I P LY I N G A N D D I V I D I N G R AT I O N A L E X P R E S S I O N S
Let a, b, c, and d be nonzero polynomials. TO MULTIPLY, TO DIVIDE,
Study Tip When multiplying, you usually factor as far as possible to identify all common factors. Note, however, that you do not need to write the prime factorizations of 24 and 60 in Example 1, if you recognize 12 as their greatest common factor.
Multiply numerators and denominators. Factor, and divide out common factors. Simplified form
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Study Tip When you multiply the numerators and the denominators, leave the products in factored form. At the very end, you may multiply the remaining factors or you may leave your answer in factored form, as in Example 2.
Visit our Web site www.mcdougallittell.com for extra examples.
Multiply by reciprocal.
= 2 = 2
11.5 Multiplying and Dividing Rational Expressions
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L AL I
USING RATIONAL MODELS IN REAL LIFE
Writing and Using a Rational Model
The models below can be created using data collected by the National Automobile Dealers Association in the United States. Five-year intervals from 1975–1995 were used. Let t represent the number of years since 1975. Number of new-car dealerships:
30,000 + 300t 1 + 0.03t
Total sales (in billions of dollars) of new-car dealerships:
80 + 10t 1 º 0.02t
a. Find a model for the average sales per new-car dealership. b. Use the model to predict the average sales in 2005. SOLUTION PROBLEM SOLVING STRATEGY
Average sales Total sales of Number of = ÷ per dealership dealerships dealerships
Average sales per dealership = A 80 + 10t
Total sales of dealerships = 1 º 0.02t 30,000 + 300t
The model predicts that the average sales in 2005 will be about $0.0463 billion. Because 1 billion is 1000 million, you can express $0.0463 billion as 0.0463 • 1000 million, or $46.3 million.
Chapter 11 Rational Equations and Functions
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GUIDED PRACTICE Concept Check
1. Describe the steps used to multiply two rational expressions. 2. Describe the steps used to divide two rational expressions. 3. ERROR ANALYSIS Describe
x + 3 4x = ÷ x º 3 x2 º 9 4x x + 3 4x • = x º 3 (x + 3)(x º 3) (x º 3)2
the error in the problem at the right. Then do the division correctly.
Simplify the expression.
3x 4x 3 4. 2 • 4 8x 3x
2x x2 º 1 5. • 3x º 3 x
x xº5 • 6. x 2 º 25 x + 5
3x • (x + 3) 7. x 2 º 2x º 15
x 2x 8. ÷ 8 º 2x 4ºx
4x 2 º 25 9. ÷ (2x º 5) 4x
x 2 º 4x + 3 xº1 10. ÷ 2x 2
3x + 1 9x 2 + 6x + 1 11. ÷ x+5 x 2 + 5x
PRACTICE AND APPLICATIONS STUDENT HELP
SIMPLIFYING EXPRESSIONS Simplify the expression.
Extra Practice to help you master skills is on p. 807.
4x 1 12. • 3 x
9x 2 8 13. • 4 18x
7x 2 12x 2 14. • 6x 2x
4x 2 16x 2 15. ÷ 16 x 8x
5x 25x 2 16. ÷ 10x 10 x
x3 13x 4 17. ÷ 7x 7x
24 5 º 2x 18. • 10 º 4x º2
xº3 4x • 19. x 2 º 9 8x 2 + 12x
º3 xº4 20. • x º 4 12(x º 7)
3x 2 9x 3 21. ÷ 10 25
x x+5 22. ÷ x+2 x+2
5x + 15 x +3 23. ÷ 3x 9x
2(x + 2) 4(x º 2) 24. ÷ 5(x º 3) 5x º 15
x2 º 36 25. ÷ (x º 6) º5x2
8 26. • (8 + 12x) 2 + 3x
Examples 1–5: Exs. 12–34 Example 6: Exs. 35–40
3x xº6 • 27. 2 x º 2x º 24 6x 2 + 9x
x • (3x º 4) 28. 2 3x + 2x º 8
5 x+1 29. ÷ x(x º 3) x 3(3 º x)
1 30. (4x 2 + x º 3) • (4x + 3)(x º 1)
x 2 º 8x + 15 31. ÷ (3x º 15) x 2 º 3x
6x 2 + 7x º 33 32. ÷ (6x º 11) x+4
x x2 x + 2 33. • ÷ 30 2 5
2x 2 5 6x 2 34. • ÷ 3 x 25
11.5 Multiplying and Dividing Rational Expressions
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RAILROAD TRAVEL In Exercises 35–37, the models are based on data about train travel from 1990 to 1996 in the United States. Let t represent the number of years since 1990. Source: Statistical Abstract of the United States Miles (in millions) traveled by passengers:
6300 º 800t 1 º 0.12t
Passengers (in millions) who traveled by train:
222 º 24t 10 º t
35. Find a model for the average number of miles traveled per passenger. 36. Use the model found in Exercise 35 to estimate the average number of miles
traveled per passenger in 1995. 37. Use the model to predict the average number of miles traveled per passenger
L AL I
SERVICE INDUSTRY CAREERS The service
industry includes a wide range of careers. Fields of service include health care, automobile and other repair services, legal assistance, education, and recreation.
in 2005. SERVICE INDUSTRY In Exercises 38–41, the models below are based on data collected by the Bureau of Economic Analysis from 1990 to 1997 in the United States. Let t represent the number of years since 1990. Total sales (in billions of dollars) of services:
NE ER T
Total sales (in billions of dollars) of hotel services:
1055 + 23t 1 º 0.04t
S = 46 + 0.7t 1 º 0.04t
www.mcdougallittell.com Total sales (in billions of dollars) of auto repair services:
48 º t 1 º 0.06t
38. Find the total sales given by each model in 1990. 39. Find a model for the ratio of hotel service sales to total service industry sales.
Was this ratio increasing or decreasing from 1990 to 1997? Explain. 40. Find a model for the ratio of auto service sales to total service industry sales.
Was this ratio increasing or decreasing from 1990 to 1997? Explain. 41.
Writing What do your answers in Exercises 38 and 39 tell you about how the sales of the service industry were changing in the period from 1990 to 1997?
PROOF In Exercises 42 and 43, use the proof shown below. Statement
ac a ? = • bc b ?
a ? = •
b a b
Explanation 1. Apply the rule for multiplying rational expressions. 2. Any nonzero number divided by itself is 1. 3. Any nonzero number multiplied by 1 is itself.
42. LOGICAL REASONING Copy and complete the proof to show why you can
divide out common factors. 2 2x º 4 43. Use the method from Exercise 42 to show that = . x+2 x2 º 4 674
Chapter 11 Rational Equations and Functions
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44. MULTIPLE CHOICE Which of the following represents the expression (x º 2)2 x 2 º 3x in simplified form? • 2x x 2 º 5x + 6 A ¡ D ¡
x(x º 3) 2 x(x º 3) xº2
x B ¡ 2
xº2 C ¡ 2
x º 4x + 4 E ¡ xº2
x2 + x 45. MULTIPLE CHOICE Which product equals the quotient (2x + 2) ÷ ? 4
1 x2 + x • 2x + 2 4
2x + 2 x + x B • ¡ 4 1
4 2x + 2 • 1 x2 + x
4 2x + 2 E • ¡ 2x + 2 x 2 + x
1 4 • 2x + 2 x 2 + x
INDEPENDENT EVENTS In Exercises 46–47, use the following information.
Two events are independent if the probability that one event will occur is not affected by whether or not the other event occurs. For independent events A and B, the probability that A and B will occur equals the probability of A times the probability of B. For example, if you draw a marble from the jar at the right, put it back, and then draw another one, the 3 3 5 5
probability that both marbles are red is • = . 46. A bag contains n marbles. There are r blue marbles and the rest of the
marbles are yellow. Find the probability of drawing a yellow marble followed by a blue marble if the first one is put back before drawing again. EXTRA CHALLENGE
47. Look back at the carnival game in Exercises 32–34 on page 668. Find the
probability of hitting the target two times in a row.
MIXED REVIEW FINDING THE LCD Find the least common denominator. (Skills Review, pp. 781–783)
3 2 48. , 4 5
2 3 49. , 9 18
1 9 50. , 16 20
14 31 51. , 54 81
QUADRATIC FORMULA Solve the equation. (Review 9.5) 52. 2x 2 + 12x º 6 = 0
COMPOUND INTEREST After two years, an investment of $1000 compounded annually at an interest rate r will grow to the amount 1000(1 + r)2 in dollars. Write this product as a trinomial. (Review 10.3) 11.5 Multiplying and Dividing Rational Expressions